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>.t). Letting h ( r ) " - #(p >~ r) for r > 0, we arrive at
poh(t + m) <. # ( f 2 ) # ( p < t) <~ m-2a(r)[h(t) - h(t + m)] + r,
r>0.
Hence,
m-2a(r)h(t) + r _2 a Po + m (r) '
h(t + m) ~
r > O.
By letting r = spoh(t) we obtain -
fh(t) Jh(t+l)
ds
-
h(t
+
1) -
(1 -- e)poh(t) h(t)
<<. -
2
PO + m - a(epoh(t))
"
Since h(t) is decreasing in t and (p0 + m-2a(epos))/s is decreasing in s, we arrive at
fh(t)
1 <<.PO + m-2a(epoh(t)) ft+ PO + m-2a(epos) (1 - e)poh(t) m dh(s) ~< Jh(t+m) (1 -- e)pOS ds. Thus, h(to)
Cm(h(to + mn))
-
Jh(to+mn)
Po + m-2a(gpos) ds >/n. (1 - e)pos W1
Therefore, #(p >~to + ran) - h(to + ran) ~ Cm1(n). C o r o l l a r y 4.2.2 In the situation of Theorem 4.2.1. (1) If a(r) - cr -5 for some c,5 > 0 and all r E (0, 1], then #(p > N) c' (logN N ) -21~ for some c' > 0 and all N > 1.
(2)//~(r)
- c[log r-l] a for some c, 5 > 0 and all r ~ (0, 1/2], then
#(p > N) <. Cl exp[-c2N1/(l+6)], for
some
c1~ c2 ~ O.
N>~ I
4.2
Concentration of measures
Pro@
191
We need only to prove for big N. If c~(r) - cr -~, then
fl--p0 p o + m - c (2e p o s ) --5as
era(t)-
(1 - e)pos
( 1 ftl-P~ s-(l+~)ds + logt -1 )
~< Cl ~
<~ c2(m-2t -5 + logt -1)
for some Cl, c2 > 0 and all t ~< (1 - p 0 ) / 2 . Therefore, " - inf{t > 0" era(t) ~< n} ~< c3(e - n V m -2/5)
r
for some constant c3 > 0 and large n. Thus the first assertion follows by taking 2 n the integer part of ~ log N. If c~(r) - c[log r - l ] 5 for some c > 0 and all r e (0, 1/2], then the desired estimate follows from Theorem 4.2.1 by noting that e l ( t ) ~ el
f l-po s-l[log Jt
s-1]Sds ~< c2(logt-1) 1+5 D
for some (71,c2 > 0 and small t > 0.
The following example shows that estimates given in Corollary 4.2.2 are sharp. E x a m p l e 4.2.1
d 2 d Consider L = d x 2 ~b(x)-~x on [0, oc) with reflecting bound-
dry at 0. Let V(x) -
/0 x b(s)ds.
It is well-known that L is symmetric with
respect to # ( d x ) " - eV(X)dx. One has C ( f , f ) " (1) Let V(x) "- - p [ l o g ( l + x ) ]
1
-~L f 2 - f L f - (f,)2.
for s o m e p > 1, i.e. b(x) -
l +- Px
then
(4.1.16) holds for c~(r) - cr -2~(p-l) for some c > 0. Thus, Corollary 4.2.2 (1) implies the exact main order of #(p > N) as N ~ oc. (2) Let V(x) "- - ( l + x ) 5 for some 5 E (0, 1], i.e. b(x) - - 5 ( l + x ) 5-1. Then (4.1.16) holds for c~(r) e[logr-1] (1-5)/5 for some c > 0 and all r E (0, 1/2] and hence Corollary 4.2.2 (2) provides the sharp main order for the decay of
-
#(p > N). In order to prove Example 4.2.1, we present below a general result concerning the estimate of c~(r) for one-dimensional diffusion processes. P r o p o s i t i o n 4.2.3
V(x) dx For a, b c C[0, co) with a > 0 such that Z ' - fo cc ea(x)
< oc, where V ( x ) " -
fo x b(r) e V(x) a(r) dr, let # ( d x ) " - Za(x----~dx and F(I, f ) " - alf'l 2.
192
Chapter 4
Weak Poica% Inequalities and Convergence of Semigroups
For any R > O, let h(R) := #((R, oc)) and 5(R) "= sup fo 9 e-V(y)dy fyn eV(y) dy x~[0,R] a(y)
Then (4.2.1) holds for a ( r ) "-- 45
Proof.
h - l ( r / ( 1 + r)),
o
r>0.
By Theorem 1.4.4, we have
)~i([0, R]) ~>
aS(R)'
d2 b d where )il ([0, R]) is the first Neumann eigenvalue of L "- a~-/x2 + dx on [0, R]. Then #(f21[o,R]) <~ 45(R)#(lVf121[o,n]),
# ( f l [ o , R ] ) - 0.
Hence the desired result follows from Theorem 4.3.1 below.
Proof of Example ~.2.1.
(1) For a - 1 and V(x) = -p[log(1 + x)], we have
5(R) <. sup
(1 + x) 2
x~[0,R] p 2 _ 1
h(R) - ( p - 1)
f
(1 + R) 2
p2 _ 1
(1 + x)-Pdx - (1 + R) 1-p
By Proposition 4.2.3, (4.2.1) holds for 4 1 2/(p-1) ~(r) -- 45 o h -1 (r/(1 + r)) ~< p2 _ i (1 + r - ) . (2) Let V(x) - - ( 1 + x) ~, 1 ~> 5 > 0. We have 5(R) -
sup ~[0,R]
/0 x e (l+s)~ds
e-(l+s) ~ds
~< c1(1 + R) 1-~ for some Cl > 0 and all R > 0. Moreover,
1 /R ~ e -(1 +s)5 ds ~< c2 (1 + R) 1-~ e -(I+R)~ h(R) - -~ <~ c3e
-(1+R)5/2
Ill
4.3
Criteria of weak Poincar~ inequalities
193
for some ca, c3 > 0 and all R > 0. Then there exists C4 > 0 such that h - l ( r ) ~ c4(logr-1) 1/5,
0 < r ~ 1/2.
Therefore, 5 o h - l ( r / ( 1 + r)) <<.c(log(1 + r - l ) ) (1-5)/5 for some c > 0 and all r E (0, 1]. Then the proof is completed by Proposition 4.2.3. [:] Finally, let us consider the general setting without assuming (4.2.2). T h e o r e m 4.2.4 Let (8, ~ ( 8 ) ) be a conservative symmetric Dirichlet form on L2(#) for a probability measure #. For any p E ~ ( L s ) with LE(p) < 1, (4.2.1) implies #(p >i to + N)
(
1
inf r +
~(p < to) ~>0
2 a(r) )
~-~
Proof. Let PN := ( p - t o ) + / N . Since r from Lemma 1.2.4 that
to N > 0
'
'
( r - t o ) + / N is convex, it follows
E (pN A 1, pN A 1) <~ 8 (pN, pN) <~
2
N 2"
By combining this with (4.2.1) for f "- PN A 1 - #(PN A 1) we obtain #((PN A 1) 2) < r + 2 N - 2 a ( r ) + #((PN A 1)2)#(p >~ to).
This implies the desired assertion by noting that #(p >~ to § N) <~ #((PN A 1)2). D
4.3
Criteria
of weak
Poincar@
inequalities
We first present a general criterion for the weak Poincar6 inequality (4.2.1) which applies in many cases. Then we go to estimate the function a for diffusions on a Riemannian manifold. To prove (4.2.1), we assume the following local Poincar6 inequality (A4.3.1) Let # be a probability measure and let L generate a conservative symmetric Markov semigroup Pt on L2(p). For any c c (0, 1), there exist A E ~,~ and c > 0 such that p(A) >~ 1 - s and #(f21A) < c S ( f , f ) + # ( f l A ) 2 / # ( A ) ,
T h e o r e m 4.3.1
f e ~(L).
(4.3.1)
If (A4.3.1) holds then
p(f2) < ~ ( r ) E ( f , f ) +
r]]fl]~,
r > 0, f e ~(L), # ( f ) = 0,
(4.3.2)
194
Chapter 4
Weak Poicar~ Inequalities and Convergence of Semigroups
where a ( r ) "= inf {c > 0 : (4.3.1) holds for some A E ~,~ with #(A) >1
1}
r+l " Proof. For any ~ E (0, 1), let c > 0 and A E ~ be such that #(A) >i 1 - ~ and (4.3.1) holds. For f E ~ ( L ) with # ( f ) = 0, one has #(/1A) 2 -- #(flA~) 2 <~ ~211f112~ . Then
(f2)
(f21A ) + llfll 2 < c (f, f) + #(flA)2 2 #(A) + ~llfll~ <~ c E ( f , f ) + 1 - e IJfll 2~ '
f e ~(L) #(f)-
O.
The proof is completed by taking s - r/(1 + r) for r > 0.
D
In the remainder of this section, we consider symmetric diffusion semigroups on a connected noncompact Riemannian manifold M of dimension d. Let d# "- e Ydx be a probability measure on M with V a locally bounded function, and 8 ( f , f ) " - #(IVfl 2) for f e C ~ ( M ) . Below we use C ~ ( M ) to replace ~ ( L ) in (4.3.2). Let o e M be fixed, and denote by p(x) the Riemannian distance between x and o. Let B~ = {p ~< r} for r > 0. To obtain explicit estimates for a, one needs to estimate the local Poincar~ constant c in (4.3.1). This is related to the first Neumann eigenvalue on a regular domain, see w T h e o r e m 4.3.2 Assume that Br is convex for any r > O. Let K E C(0, c~) be a nonnegative increasing function such that the Ricci curvature on Br is bounded below b y - g ( r ) . Then (4.3.2) holds with a ( r ) - 4R~ cosh d-1 [ R r v / K ( R r ) / ( d 7r2 V/1 + 8R2rK(Rr)/Tr 4
1)] exp[SR~(V)],
(4.3.3)
where Rr := inf{s > 0: #(B~) ~< r / ( l + r ) } , S R ( V ) := s u p { V ( x ) - V ( y ) : x , y e B R}. Where for d = 1 we put g = K / ( d - 1) = 0. If V E C2(M), let K v be an increasing function such that Ric-Hessy is bounded below b y - g y ( r ) on Br. Then (4.3.2) holds with exp [~Kv(Rr)R2r] - I a(r)
Proof.
-
gy(
)
(4.3.4)
"
By (2.2.8), one has
A(R) " - inf {#(IVfl21BR) " f e C I ( B R ) , # ( f l B R ) - 0, #(f21sR) -- 1} 7r 2
-~-~V/1 + 8 R 2 K ( R ) / ~ 4 cosh 1-d [ R v / K ( R ) / ( d -
1)], (4.3.5)
4.3
Criteria of weak Poincar6 inequalities
195
where for d = 1 we have K = 0 and set K / ( d - 1 ) = 0. Then, b y a s i m p l e comparison argument, we see (4.3.1) holds for A = BR and c-
4R2 c~ [Rv/K(R)/(d7r2 V/1 + 8R2K(R)/Tr 4
1)] exp[hR(V)].
Therefore, the first assertion follows from Theorem 4.3.1. For the case where V E C2(M), by (2.2.2), (4.3.1) holds for A
BR and
exp [ ~ K y ( R ) R 2] - 1 C=
Kv(n)
which proves the second assertion.
D
It is clear that the convexity assumption on B~ in Theorem 4.3.2 was made to use known estimates for the first Neumann eigenvalue. This assumption is however not true in general. To treat the general case, we present the following result obtained by perturbation. Theorem
4.3.3
Assume that there exist ro > 0 and V E C[r0, c~) such that Lop := (A + V Y ) p <. V(P)
on Pro in the distribution sense.
~
(4.3.6)
For any ~ > 0 and R >~ ro, let uc(R) =
R(c+v(r))+dr. If there exists ~ > 0 such that # ( B ~ ) e x p [ ~ ( R + l ) - r l ~ ( R ) ] --~ o
0 as R ~ c~, then there exists c(c) > 0 such that (4.3.2) holds with a(r) = c(e) e x p [ ~ ( / ~ + 1)], where / ~ "-- inf { R >~ ro " # ( B R ) - I + c ( ~ ) e x p [ u ~ ( R + I ) - ~ ( R ) ] <~ r / p ( B ~ ) } ,
r > O.
Especially, if r/~(oc) < oc for some ~ > O, then the Poincard inequality (1.1.4) holds for C = c(r Proof. Let L - L0 - l{p>~0)(~ + V(p))+Vp, where Vp(x) "= 0 if x is in the c cut locus of o. Then Lp <~ - c on B~0 in the distribution sense. By Corollary 1.4.13, there exists c1(~) > 0 such that r'(f 2) - r,(/) 2 < C1(c)r,(lVf] 2)
(4.3.7)
for all Lipschitz continuous f C L2(#), where d~ "= C exp[-l{p>.~o)~(p)]d # with C > 0 a normalizing constant. For f c C ~ ( M ) with # ( f ) = 0 and R > r0, let h = ( R + 1 - p)+ A 1. We have # ( f 2 h 2 ) - # ( f h ) 2 / # ( g R + l ) - ~nf # ( ( f h -
r i B s + i ) 2)
~< C -1 exp[u~(R + 1)] inf y ( ( f h rCIR
rlsR+~) 2)
196
Chapter 4
Weak Poicar6 Inequalities and Convergence of Semigroups
-- C -1 exp[~le(R + 1)] (v(f2h 2) - ~,(fh)2/v(BR+l)) ~< C -1 exp[~e(R + 1)] (~(f2h2) - ~(fh)2). Combining this with (4.3.7), we obtain (recalling that #(f) - 0) #(f2h2) < #(fh)2/#(BR) + C-lcl (e)exp[~e(R + 1)],([V(fh)[ 2)
< [[f[[2#(B~)2/#(BR) + 2c1(e)exp[r/~(R + 1)]#([Vf[ 2) + 2c1(e)exp[rk(R + 1) - ~(R)]#(B~)[[f[[~ for some Cl(e) :> 0. This proves the theorem for c(e) - 2c1(e) since #(f2) ~< # ( f 2h2) + IlfllL~(s~). T h e o r e m 4.3.4 Assume that ~/ in (4.3.6) is negative and B~ is convex for any r > O. For any R >~ro, let r
:=
inf [-~/(r)], re[to,R]
r
:= inf{s ~> r0" r
2 ~> 9#(BsC)},
l/le
cos l D
Then there exists c > 0 such that (4.3.2) holds for a(r) = c/((R~) provided / ~ "- inf {R >~ro " #(B~_I)[I+c/((R)] <. r # ( B R _ l ) / ( r + l ) } < c~,
r>0.
Proof. By Theorem 4.3.3, the Poinca% inequality (1.1.4) holds provided r > 0. Hence we only consider the case where r - 0. In this case one has r --~ oc as R ~ oc. Then there exist e > 0 and R(e) ~> r0 + 1 such that r162162
- ro) 2 ~ 8(1 + e)#(B~(R)),
Next, since Lop <~ - r 2.5.4 we obtain
R i> R(e).
(4.3.8)
in BR \ Bro in the distribution sense, by Corollary
A0(BR \ B~o)"- inf (#(IVfl2 9 f e C ~ ( B R \ B~0), #(f2) _ 1}/>
r
~ ~
4 (4.3.9)
Next, the proof of [194, Theorem 1.1 ] yields that A(R) " - i n f {#(IVfl2) 9 f e C ~ ( B R ) , # ( f 2) - # ( f ) 2 _ 1} A0(BR \ Bro)A(r)#(Br)(r - ro) 2 - 2A(r)#(Brc) ) 2A(r)(r - r0) 2 + A0(BR \ B~o)(r - ro)2#(B~) + 2#(B~)'
r ~ (to, R).
Combining this with (4.3.5), (4.3.8) and (4.3.9), we obtain - r0) 2
ClA(r162162162
-
~(R) t>
A(r162
-
r0) 2 + r162
-
ro)2#(Br
+ #(Be(R) )
4.3
Criteria of weak Poincar4 inequalities
197
for some Cl, c2 > 0 and all R ~ R(~) such that r < R. If r >~ R, then A(R)(~> A(R)) >~ c2((R) still holds for some c2 > 0 according to (4.3.5). Then for any f c C ~ ( M ) with # ( f ) - 0 and any R >~ R~, let h - ( R - p)+ A 1, we have #(f2) ~< #(f2h2 ) + I l f l l i # ( B ~ - l )
2#(iVf[2) + 21[fll c~#(BR-1) 2 ~ c2~(R)
2 ~ [[f lI~#(BR-1) +
#(BR-1)
"
This proves the desired result by taking c - 2/c2 for small r > 0 such that D
Finally, we present the following example with Pt having some typical convergence rates, which extend Example 4.2.1 to high dimensions. E x a m p l e 4.3.1
Let E = Rd, L = A + V V , # ( d x ) -
I
Ff
"i
eY(X)dx/~/eY(X)dx],
and E ( f , f) - #(IVfl 2) for the choices of V specified below. Let ~ ( f ) - [[fll~. (1) For p > 0, let Y(x) - - ( d + p)log(1 + Ixl) and ~- - min{(d + p + 2)/p, (4p+4+2d)/[p2-4-2d-2p]+}. Then (4.2.1) holds with a ( r ) - c ( l + r -~) for some c > 0, and there exists c' > 0 such that
#((Ptf) 2) <
c'llfll~t -~/~,
t >~0, # ( f ) -
0.
(4.3.10)
(2) Let p > 1 and V(x) - - d l o g ( 1 + [ x l ) - ploglog(e + [xl). Then (4.2.1) holds with c~(r) - Cl exp[c2r -1/(p-1)] for some Cl, c2 > 0, and there exists c > 0 such that
#((Ptf) 2) <~all filL[log(1
+ t)] l-p,
t > 0, # ( f ) -- 0.
(4.3.11)
(3) Let V(x) - -crlx[ 5 for some a, 5 > 0. It is well-known that the Poincar4 inequality holds if and only if 5 >~ 1, so we only consider the case 5 E (0, 1). For 5 c (0, 1), there exist C, Cl,C2 > 0 such that (1.6) holds with a ( r ) c[1 + log(1 + r - l ) ] 4(1-5)/5, and
P((Ptf) 2) <~cxllflli exp [- c2t~/(4-35)],
# ( f ) - 0, t >~ 0.
(4.3.12)
Proof.
By Corollary 4.1.5, it suffices to prove (4.2.1) with a as specified there. We note that K - 0 since M - R d. Let R~ and /~r be defined in Theorems 4.3.2 and 4.3.4 respectively. (a) In this case we have 5 R ( V ) - (d + p)log(1 + R) and Rr <~ c(1 + r -1/p) for some c > 0 and any r > 0. By Theorem 4.3.2, (4.3.2) holds with
4R~ (d+p+2)/p a(r) - ~ exp[6R~(V)]<~clr-
(4.3.13)
198
Chapter 4
Weak Poica% Inequalities and Convergence of Semigroups
for some Cl > 0 and all r e (0, 1] (hence for all r > 0 since #(f2) ~< IIfll )Next, assume that p 2 - 4 - 2 d - 2 p > 0. Let r0 ~> 1 be such that ( d - 1 ) / r o - ( d + p)/(1 +r0) < 0. It is easy to see that r ~> c2R -1, r <~ c3R 2/(2+p) , ((R) c4R -2(d+2+2p)/(2+p) for some c2, c3, c4 > 0 and all R ~> to. Moreover, we have ~ c5r-(2+p)/(p2-2d-4-2p) for some c5 > 0 and all r E (0, 1]. Therefore, by Theorem 4.3.4, (4.3.2) holds with -
<
for some c, c6 > 0. Combining this with (4.3.13) we prove (4.3.2) for a ( r ) = c7 r - T for some c7 > 0. (b) Obviously, 5R(V) = d log(1 + R) + p log log(e + R), and Rr ~< exp[c(1 + r-1/(p-1))] for some c > 0. By Theorem 4.3.2, (4.3.2) holds with
-
4R2r exp[6R,(V)] <. Cl exp[c2r-1/(p-1)]
for s o m e c1~c2 ~ 0. (c) In this case Theorem 4.3.4 provides better result than Theorem 4.3.2. Let 5 E (0, 1). We have v(r) -
d- 1 r
a5
rl_~ , which is negative for big r.
Taking r0 > 1 such that 7(r0) < 0. We see that r ~> clR ~-1 for some cl > 0 and all R ~> r0. It is easy to see that there exists c2 > 0 such that #(B~) <<.c2 exp[-as~]s 1-~ for s >~ r0. Let SR > 0 solve 9c2 exp[--as~]s -(1+~) -c2R 2(~-1), then r ~< sn V ro. Hence exp[5r
(V)] - exp[ar
~] ~< exp[a(sR V r0) 5] ~< c3R 2(1-~)
for some C3 > 0. Since K - 0, ~(R) - r
2 exp[-5r
(V)] /> c4R 4(~-1)
for some c4 > 0. Then # ( B ~ _ I ) ~ ( R ) -1 ~< c5 exp[-aR~/2] for some c5 > 0 and all R > r0. We o b t a i n / ~ ~< c6(1 + [log(1 + r - l ) ] 1/5) for some c6 > 0. Therefore, by Theorem 4.3.4, (4.3.2) holds for
a(r) = cr
exp[5r
~ c7[log(1 + r - l ) ] 4(1-5)/5
for some c7 > 0 and all r e (0, 1] (hence all r > 0). On the other hand, by (4.3.3) one obtains (4.3.2) for a ( r ) - c(1 + r -~) for some c, e > 0. This choice of c~ is worse than the one above. K]
4.4
4.4
Isoperimetric inequalities
199
Isoperimetric inequalities
The aim of this section is to prove (4.2.1) using isoperimetric inequalities for diffusions on a manifold and symmetric jump processes. We only consider the ergodic case, but similar results hold for the case where p is infinite, see [205] for details. 4.4.1
Diffusion processes on manifolds
Let M be a connected noncompact Riemannian manifold, and # a probability measure on M. Define
k(r) "-
inf
tt(A)E[r,1/2]
po(OA) #(A) '
r e (0, 1/2],
(4.4.1)
where A runs over all open smooth domains (according to [224], we may also assume that A is connected), and #o(OA) denotes the area of OA induced by #. Theorem
4.4.1
If k(r) > 0 for any r e (0, 1/2], then r > O, f e C ~ ( M ) , # ( f ) -0,
#(f2) ~< a ( r ) p ( i V fl2) + r S . ( f ) 2
(4.4.2)
inf s u p { f ( x ) - f ( y ) " x , y e A}. In #(A)--1 particular, if k(O) "- lira k(r) > O, then (1.1.4) holds for C - 4/k(0) 2. where a(r) = 4k(r/2) -2 and 5 . ( f ) r---~O
Proof. Assume that k(r) > 0 for r E (0, 1/2]. Let f E C ~ ( M ) be such that # ( f ) - 0 . Take ro c [inf f, sup f] such that # ( f > ro) V # ( f < ro) <~ 1/2. For s > 0, let t~ "-- inf {t >~ 0" # ( ( f - r0) +2 > t) >~ s}. By (4.4.1) and the coarea formula, we obtain
-ro)+l L
#((f-
ro) +2) -<
foll(f
#((f-
ro) +2 > t)dt
# o ( ( f - r o ) +2 k(s)
f~ 1
~< k ( s ) # ( I V ( f - to)
-t)
+2
[) +
dt + sll(f - ro
)+ I1~2
~,
sll(f
-
ro)+ll 2
s>O.
The same estimate holds for (f - r0)- in place of (f - r0) +, Then 1
+2
~"(s)~ + k(~)~(Iv(f- ~o) I+ Iv(s-~o) >~ # ( ( f
-
ro) +2
+
(f
-
ro) -2)
-
#((f
-
ro)2).
_2
I)
(4.4.3)
200
Chapter 4
Weak Poicar6 Inequalities and Convergence of Semigroups
Noting that
# ( I V ( f - ro)+21+lV(f - to)-2 l) - ~(IV(f-ro)21) < 2v/~(lVfl2)~((f - to)2), by (4.4.3) we obtain 4 #(f2) ~ # ( ( f _ ro)2)< k(s)2#(lV f 12) q- 2s6~(f) 2,
s>O.
This implies (4.4.2) by taking s -- r/2. T h e o r e m 4.4.2 Assume that V E C2(M) such that d# "- eVdx is a probability measure and that IVPtfl 2 <. h(t)PtlV fl 2 holds for some positive h E C[0, c~) and all t > O,f E C ~ ( M ) , where Pt is generated by A + V V on L2(#). /f (4.4.2) holds, then for any e e (0,1/2) there exists c(e) > 0 such that k(r) >I c(e)/a(er).
Proof. We first note that the assumed gradient estimate implies Ptl = 1. Moreover, this assumption implies ds IVPtfl2 9- c ( t ) l V p t f l 2 P t f 2 - (Ptf) 2 ) 2 Jot h(s)
f e C~(M).
Then
IIVPtfll~ < I l f l t ~ v / 1 / c ( t ) ,
t>O.
Hence for any smooth g with Ilgll~ ~ 1,
#(g(f - Ptf)) - -
#(g(A + V V ) P s f ) d s -
~< ~(IVfl)
#((VPsg, V f ) ) d s
IIVGgll~cds <<.ct~(IV f[)
for some c > 0 and all t ~> 1. Therefore,
# ( I f - Ptfl) ~ ct~(IV f[),
t/> 1.
(4.4.4)
For any r e (0, 1/2] and any smooth domain A with #(A) e [r, 1/2]. Take {fn} C C ~ ( M ) s u c h that fnlA = 1, f n ( X ) - 0 if dist(x, A) >~ 1/n, and IVfnl <~ n + 1/n. Applying (4.4.4) to fn and letting n ~ c~, we arrive at
ct#o(OA) >1 ~(iA(i -- PtiA)) + ~(iA~PtiA) - 2[#(A) - ~(iAPtiA)] = 2#(A)
- 2#((Pt/21A)2).
(4.4.5)
4.4
Isoperimetric inequalities
201
If (4.4.2) holds, by Theorem 4.1.3 we have
#((Pt/21A) 2) <~ s + exp[-t/a(s)]#(A) + #(A) 2,
s>0.
Therefore, (4.4.5) implies
#o(OA) #(A)
2
>~ ~,t>0sup~ { 1 - s/#(A) - exp[-t/a(s)] - #(A)}
1 - 2 s / r - 2exp[-t/a(s)] s,t>o ct
~> sup
we obtain For any e e (0,1/2), taking s = er and t = a ( e r ) l o g ~ , 1 - 2~
#o(OA) c(e) p(A) >~ a(r for some c(c) > O.
[:]
C o r o l l a r y 4.4.3 Consider the situation of Theorem 4.4.2. If (4.4.2) holds then for any c e (0, 1/2) there exists c(e) > 0 such that
~ Proof.
Let h ( s ) =
1 a(er) dr >/ c(g)R. (p~>n) r
(4.4.6)
#(p >~ s). By Theorem 4.4.2, (4.4.2)implies that
_h'(s)a(cs) >~ c(~). This proves (4.4.6). h(s)
K]
Obviously, for a given function c~, (4.4.6) provides an estimate of the decay of #(p >~ R) as R ~ c~. In particular, if (1.1.4) holds then by (2.4.3) there exists c > 0 such that #(exp[cp]) < c~. This is already known by the concentration of measures for Poinca% inequalities. Next, let us consider Dirichlet forms on ]~d with the property. Let # be a probability measure on ]~d and E(f, f ) " - #((aVf, V f ) ) , f e C ~ ( M ) , where a(x) - (aij(X))d• is positive definite for any x E I~d. Let r r be two positive continuous function such that
(~l(X)[y[ 2 ~ (a(x)y,y} ~ r
2,
x,y C ]~d.
Finally, let d#0 " - r and d~_v " - r d#0 be defined on the boundary of any smooth domain, where d#0 is induced by # and the standard Euclidean metric. The proofs of Theorems 4.4.1 and 4.4.2 imply the following result. m
Theorem 4.4.4
by #o ( resp. p_~) .
Let k(r)(resp, k_(r)) be defined in (4.4.1) with #o replaced
202
Chapter 4
Weak Poicar6 Inequalities and Convergence of Semigroups
(1) If k_(r) > 0 for r e (0, 1/2], then #(f2) < ~ ( r ) # ( ( a V f , V f)) + r 6 z ( f ) 2,
f e C ~ ( R d ) , # ( f ) - O, r > 0 (4.4.7)
for a(r) - 4_k(r/2) -2. (2) Assume that aid E C2(Rd),d# - eYdx for some V E C2(Rd), and that there exist K >~ 1 and a matrix-valued function a such that K - 1 I ~ a ca* ~ K I and s u p ix - yl-2[DLo(x) - r x#y
d
where
bi - E
0
+
(4.4.8)
0
[aij-~xjV q- -~xjaij]" If (4.4.7) holds then for any e E (0, 1/2)
j=l
there exists c(e) > 0 such that k(r) ) c(e)/a(er). Proof. d
Let Pt be the Markov semigroup generated by the closure of 02
d
0
bi-d-~.. One has ( a V P t f , V P t f ) < K2e2Ktpt (aV f , V f}
y~. aij OxiOxj -}- E i=1
i,j--1
for f e C~(]Rd), see (9.1) in [51]. Hence the proof of Theorem 4.4.2 applies to the manifold ]1~d with the metric induced by a -1. [3 The following is a simple consequence of Theorem 4.4.4 in the one dimensional case. C o r o l l a r y 4.4.5 Consider the situation of Theorem 4.4.4. Let d - 1 and d# - eYdx for some Y E C(I~). For any r E (0, 1), let Cr > 0 be such that #([-Cr, Cr]) = 1 -- r. Then
~(r)-k_(r)>~ inf !
inf
s~[~,1/2] s t ~ [ - ~ , ~ ]
v/a(x)exp[V(x)]'-a(r).
(4.4.9)
Consequently, (4.4.7) holds for a(r) - 4n(r/2) -2 provided it is finite. On the a t2 1 a'V' + 21 a" other hand, if a, Y e C2(I~) such that aV" + -~ 4a is bounded from above, then (4.4.7) implies k(r) - k_(r) >~ c(r and some c(r > O.
for any r e (0, 1/2)
Proof. In the present case we have #o(x) - #o(x) - (v/-aeY)(x). Then for any r e (0, 1/2] and connected I C ]R with #(I) - r , we have OI [~[-Cr, c~] # O. This proves (4.4.9). To prove the second assertion, we consider the metric
induced by a -1"
0 12 _ a - 1 Therefore, under this metric Ric - 0 (since
~Xa
4.4 d-
Isoperimetric inequalities
203
0
a-1/2dx which
1) and the unit vector field is X - v / a ~ xx. Next, let d 2 -
is the Riemannian volume element. Then we see that d# - exp V + ~ log a d~. Therefore, Hessv(X, X ) " -
la~v~ + -~ la. X 2 V - aV" + -~
a ~2 4a
The proof is completed by Theorem 4.4.4.
K]
Finally, we apply Corollary 4.4.5 to the first two cases in Example 4.3.1 to obtain better choices of c~ for d = 1, where the first case goes back to Example 4.2.1 (I). (1) For p > 0 and V(x) - - ( 1 + p)log(Z + Ixl), we have c~ <<.air" - l I p for some c > 0. Then tr ~> cr 1/p and hence (4.4.7) holds for a ( r ) - clr -2/p and some Cl > 0. Moreover, it is easy to see that k(r) ~< c'r 1/p for some c' > 0 and V" is bounded above. Hence, by Corollary 4.4.5, (4.4.7) does not hold for any with a(r)r 1/p --~ 0 as r --~ 0. (2) Let p > 1 and V(x) = - l o g ( 1 + I x l ) - p loglog(e + Ixl). Similarly to (1), we have ~(r) ~> cr 1/(p-1) e x p [ - r -1/(p-1)] for some c > 0 and r e (0, 1/2]. By Corollary 4.4.5, (4.4.7) holds for a ( r ) - clr -2/(p-1) exp[2P/(P-1)r-1/(p-1)] for some Cl > 0. Moreover, there exists c' > 0 such that k(r) ~< c'r 1/(p-1) exp[-r-1/(p-1)]. Hence, by Corollary 4.4.5, (4.4.7) does not hold for any with a(r)exp[--sr -1/(p-1)] --+ 0 as r -~ 0 for some s e (0, 21/(p-1)). 4.4.2
Jump
processes
Let J be a symmetric measure on (E • E, ~ • ~ ) . Define 8(f, f)'-- ~
• [f ( x ) - f (y)]2J(dx, dy),
~ 9 ( E ) - {f e ~ ( E ) " 8 ( f , f) < oc}.
We consider t h e inequality
p(f2) ~ a ( r ) 8 ( f , f ) + r 6 . ( f ) 2,
r > 0, # ( f ) - O.
(4.4.10)
If (4.4.10) holds, then for r e (0, 1/2] and A with #(A) - r e (0, 1/2], taking f - 1d in (4.4.10) we obtain
k(r) "-
inf J ( A • A c) 1 - s/r 1- s ,(A)c[r,1/2] #(A) ~> sup ~ ~> , s>0
s E (0, 1).
(4.4.11) Therefore, the main task is to prove (4.4.10) using isoperimetric inequalities. Let V be a nonnegative symmetric measurable function ~/on E • E satisfying
204
Chapter 4
Weak Poicar6 Inequalities and Convergence of Semigroups
(1.3.7) with K = 0. Define J(1/2)(dx, dy) = l{v(x,y)>O}J(dx, dy)/v/~/(x, y), and let/r be defined in (4.4.11) with J replaced by j(1/2). We have the following result. T h e o r e m 4.4.6 If k(r) > 0 for r e (0, 1/2], then (4.4.10) holds for a(r) 2k(r/2) -2. Consequently, if ~:(0) > 0 then (1.1.4) holds for C - 2k(0) -2.
Proof.
Let r e (0,1/2]. For bounded f with #(f) = 0 and 8(f, f) < oc, let r0 be such that # ( f > r0)V # ( f < r0) ~< 1/2. For any t >~ 0, let At "- { ( f - r0) +2 > t} and p ( t ) " - #(At). Then we have p(t) <. 1/2. Let t~ "- inf{t > / O ' p ( t ) <~r}, then V/28((f - ro)+, (f - ro)+)#((f - ro)+ 2)
> ~I f
f
[(f(~)-~~ + 2 -
(f(y)
-
ro)+2lJ(dx, dy)
(s(~)_~o)+>(s(y)_~o)+}[(f(x) r~ +2 (f(y)
ro)+2]j(dx, dy)
k
>1
J(At x A~)dt >>.k(r)
p(t)dt.
Therefore,
fll(f -ro)+ll~ # ( ( f -- ro) +a) =
p(t)dt Jo
1 V/2E(( f _ to)+ ' (f _ to)+)#(( f ~< ~(~)
_ ro)+2)+rll(f
_ ro)+l[ 2
for all r > 0. This implies /z((f - to) +2) ~ _ 2
k(~)2 8 ( ( f
- to) +, (f - ro) +) +
2rll(f
-
ro)+ll 2o~,
r > O,
(4.4.12) Similarly, (4.4 12) holds for ( f - r0)- in place of ( f - r0) +. Then the proof is completed by noting that #(f2) K # ( ( f _ r0)2) and I(f(x) - to) + - (f(y) - to)+12 + [(f(x) - ro)- - (f(y) - ro)-[ 2 (4.4.13) ~< I f ( x ) - f (y)l 2. Indeed, (4.4.13) is obvious when ( f ( x ) - r o ) ( f ( y ) - r0) > 0. In the case where ( f ( x ) - r o ) ( f ( y ) - ro)) <. O, we have I f ( x ) - f ( y ) [ - [ f ( x ) - ro[ + I f ( y ) - rol and hence (4.4.13) holds. E]
4.4
Isoperimetric inequalities
205
C o r o l l a r y 4.4.7 Assume that there exists R > 0 such that J ( A x E) ~< R # ( A ) , A e ~ . Taking ~/= R we obtain k(r) - k ( r ) / v ~ . Therefore, (4.4.10) holds for some a if and only if k(r) > 0 for r e (0, 1/2], and in this case (4.4.10) holds for a(r) - 2Rk(r/2) -2. C o r o l l a r y 4.4.8 Consider the birth-death process with Dirichlet form given by (1.3.5). Take ~/(i,j) "- (hi + bi) V (aj + bj). For any r e (0,1/2], let ir "-- inf{i > 0 9 E #(J) <<"r} and j >~i
p(r)
"-- inf
{ (#(i + 1)ai+1) A (#(i)bi)
0~i~ir}.
v/(ai + bi) V (ai+l + bi+l) Then k(r) ~ infs~[r,1/2]p(s)/s and (4.4.10) holds for a(r) - 2{infse[r/2,1/2] p ( s ) / s } -2 provided it is finite for any r e [0, 1]. Proof. Let r e (0, 1/2]. For any s e [r, 1/2] and I C Z+ with #(I) = s, we have [0, i~] ~ I # o, [0, i~ + 1] ~ I ~ r O. Then, for ~/(i, j) = (hi + bi) V (aj + by) we have iEI,j~I Finally, we present some examples for birth-death processes which have the same convergence rates as the ones given in Corollary 4.1.5.
E x a m p l e 4.4.1 choices of bi.
Let ai - 1 for i ~> 1. We consider the following three
(1) Let bi - ( ) 5+i 1
for some 5 > 1 and all i ~> 1. We have #(i) - #(O)i -~
for i ~> 1. Obviously, by Corollary 4.4.8 k(r) ~> cr -1/(5-1) for some c > 0, hence (4.4.10) holds for a ( r ) - c'r 2/(1-5) for some c' > 0.
_i 1 (log(1 i))hlog(2 + i)
(2) Let bi - i 4-
for some 5 > 1. We have #(i) - i - l ( l o g ( l +
i))-h#(O),i ~> 1. By Corollary 4.4.8 there exist Cl,C2 > 0 such that k(r) >~ Cl exp[-c2rl/(1-~)], and hence (4.4.10) holds for (~(r) - exp[c(1 + rl/(1-~))] for some c > 0. (3) Let bi - exp[a(i ~ - ( i + 1)~)] for some a > 0, 6 e (0, 1) and all i ) 1. We have # ( i ) - exp[-ai~]#(O), i ) 1. Since E exp[-aJh] ~< clil-~ exp[-ai~] j~i .l ~< c2[log(1 + r - l ) ] 1/6 for some for some Cl > 0 and all i ~> 1, we have i~ ~< z~ .l c2 > 0, where z~ > 0 satisfies c1(i$)1-~ exp[_a(i,)5] _ r. Then by Corollary 4.4.8, k(r) >~ c3 exp[_a(i,r)5] _ c__33(i,r)5_ 1 ~ c4[log(1 r Cl
+
r - l ) ] (5-1)/5
206
Chapter 4
Weak Poicar~ Inequalities and Convergence of Semigroups
for some c3, c4 > 0. Therefore (4.4.10) holds for a(r) - c[log(1 + r - l ) ] 2(1-5)/5 for some c > 0. Finally, it is easy to see that (1.1.4) holds if 5 >/1. In the next example we consider some birth-death processes with unbounded rates. E x a m p l e 4.4.2 Letting a~ - bi for i ~ 1, we have #(i) - a~-l#(0), i ~ 1. (1) Let ai - i ~ for some 5 > 1 and all i/> 1. Then ir ~ clr 1/(1-~) for some Cl > 0. By Corollary 4.4.8, k(r) ~> c2r(2-~)+/(2(~-1)) for some c2 > 0. Hence (4.4.10) holds for a ( r ) - c3r(2-~)+/(1-~) for some c3 > 0. Especially, if 5 >~ 2 then (1.1.4) holds. (2) Let ai = i [ l o g ( l + i ) ] ~ for some 5 > 1 and all i i> 1. Then ir < exp[clr 1/(1-~)] for some Cl > 0. By Corollary 4.4.8, there exists c2 > 0 such that k(r) >~ exp[-c2r 1/(1-5)] and (4.4.10) holds with a(r) = exp[c(l+rl/(1-5))] for some c > 0. (3) Let ai -- i2[log(1 + i)]-~ for some 5 > 0 and all i ~> 1. Then ir clr-l[log(1 + r - l ) ] ~ for some Cl > 0. By Corollary 4.4.8 fc(r) ~> c2[log(1 + r - l ) ] -~/2 for some c2 > 0 and (4.4.10) holds with a ( r ) - c[log(1 + r - l ) ] ~ for some c > 0.
4.5
Notes
This chapter is reformulated from [158] and [215]. Although in w and w we only considered the conservative case where # is a probability measure, analogous results for the other case can be found in [207]. The study of algebraic convergence was initiated by [134] using the following Nash type inequality (also called the Liggett-Stroock inequality, see e.g. [a7]): #(f2) ~ C S ( f , f ) l / p ~ ( f ) l / q
#(f)-
0, f E ~ ( L ) ,
(4.5.1)
where p,q E (1, oe) with p-1 _~_q-1 _ 1, C is a positive constant, and 9 9 L2(#) --~ [0, c~] satisfies ~ ( c f ) - c 2 ~ ( f ) for any c E N and f E L2(#). Liggett proved that if ~ ( P t f ) <~ ~ ( f ) for f E ~ ( n ) and t ) 0, then (4.5.1) implies
# ( ( P t f ) 2) <<.c ~ ( f ) t l-q,
t > 0, f E L2(#), # ( f ) - 0
(4.5.2)
for some c > 0. If moreover Pt is symmetric then (4.5.2) implies (4.5.1) for some C > 0. It is easy to see that (4.5.1) holds for some C > 0 if and only if (4.1.1) holds for ~(r) " - cr 1-p for some c > 0. Thus, Liggett's result is recovered by Corollary 4.1.5 as well as Corollary 4.1.8. Following the line of [134], the algebraic convergence of Markov chains was studied in [55].
4.5
Notes
207
The notion of weak Poincar~ inequality was introduced in [158] and further studied in [3], [4], [219], [203], [208]. This notion is also related to some earlier work. For instance, in the context of conservative Dirichlet forms, [102] proved that the weak spectral gap property is equivalent to the uniformly positivity improving property, i.e. for any ~ > 0, inf{#(1APtlB): # ( A ) , # ( B ) ~> 6} > 0, then by Proposition 4.5.1 they are also equivalent to the weak Poincar~ inequality. Moreover, the equivalence of the weak spectral gap property and the convergence of Pt was observed by [143]. See [208], [209] and references within for more equivalent statements of the weak Poincar~ inequality. Comparing with these equivalent statements, the advantage of the weak Poincar~ inequality is that it provides quantitative estimates of the convergence rates for Markov semigroups. When ~ ( f ) - I l f l l ~ , the weak Poincar~ inequality for Dirichlet forms describes the weak spectral gap property which interpolates the irreducibility of the Dirichlet form and the existence of spectral gap. There are a plentiful examples where this inequality holds but the Poincar~ inequality does not (see e.g. Example 5.2.1 and note that the Poincar~ inequality implies the exponential integrability of the distance). So, it is interesting to also distinguish W S G P from the irreducibility. This was realized in [158] which shows that the second quantization Dirichlet form of the Brownian motion o n I~ d is irreducible but does not satisfies the weak Poincar~ inequality. But so far there is no any such example in finite dimensions. Concerning the concentration of #, although an explicit estimate is presented in Theorem 4.2.4 for general conservative Dirichlet forms, but this bound is less sharp than those included in Theorem 4.2.1 and Corollary 4.2.2 for the diffusion case. In particular, when a is constant, Theorem 4.2.4 gives #(p >~N) <~cN -2 for some c > 0 and all N > 0. But in this case #(p ~> N) decays exponentially in N as N -+ c~. So, sharp concentration estimates are yet to be searched for the weak Poincar~ inequality for general conservative Dirichlet forms.
Chapter 5 Log-Sobolev Inequalities and Semigroup Properties In this chapter some contractivity (or boundedness) properties of Markov semigroups are characterized by using the log-Sobolev inequalities. Although these properties have been studied in w by using super Poincar6 inequalities, results therein and those introduced below have their own advantages. For instance, the framework in w is more general, in particular, Theorem 3.3.13 provides sharp criteria for the hyperboundedness and the superboundedness for (non-symmetric) semigroups. On the other hand, although results in w only work for the symmetric case (or a diffusion case as explained in Remark 5.1.1 below), the estimate (5.1.5) of the hyperbound is sharp and hence powerful in applications. Next, we study the existence of spectral gap for hyperbounded operators. In particular, the optimal sufficient condition on hyperbound for the existence of spectral gap is presented. The concentration of measures, the criteria of log-Sobolev inequalities and the relationship between the strong ergodicity and the hypercontractivity are also studied.
5.1
Three boundedness properties of semigroups
Let (F, ~ , #) be a a-finite measure space and (8, ~(d~)) a Dirichlet form on L2(#) associated with sub-Markov semigroup Pt and generator (L, ~(L)). We say that 8 satisfies the log-Sobolev inequality if there exist C1 > 0 and 6'2 E R such that #(f2 log f2) <~ C18(f, f) + C2,
f e ~ ( 8 ) , # ( f 2 ) _ 1.
(5.1.1)
If # is finite and d~ is conservative, then 6'2 i> 0. In this case, we call (5.1.1) the strict or defective log-Sobolev inequality with respect to C2 - 0 or 6'2 > 0. If # is infinite, (5.1.1) may hold for some C2 < 0, e.g. for the Brownian motion o n ]~d one has IIPtll2 at-d~4for some constant c > 0, where # ( d x ) : = dx
5.1
Three boundedness properties of semigroups
209
is the Lebesgue measure, then Theorem 5.1.7 below implies (5.1.6) below for g(r) < 0 if r is big. To study the hypercontractivity of Pt using (5.1.1), we first introduce some lemmas. L e m m a 5.1.1 Let ~+ "- { f ) O" f e ~ ( L ) , f, L f e L ~ ( # ) ~ L I ( # ) } . (1) For any p e (1, oc), ~+ is dense in LP+(#) "- { f e LP(#) 9 f ) 0}. (2) If 8 is symmetric, then ~+ is dense in ~ + ( oz) "- {f e ~ ( 8 ) 9 f >~ 0} under 81/2 .
Proof. (a) Since L 1(#) N L ~ ( p ) is dense in L~_(#), it suffices to show that for any f e L~_(#) ~ L ~ ( # ) there exists a sequence {fn} C ~+ such that fn --~ 1
f in LP(#). Let f C L~_( # ) ~ L~c (#) be fixed. Take fn "- n Then, by the contraction of Ps in LI(#) and L~
L~
/0; P~fds,
n >~ 1.
1 [ I f - full p <~ n
L
<~ n~ p-1
#(IPsf - flP)ds 1
/0
1
liNeN
~< 2llf[[le p-1 +
flllds +
-
n
/0n
liNer - f l l ~ ( I P ~ I
- fl > e)d~
211fll~f01 [[Psf- fll2ds. E
By the C0-property of Ps in L2(#) and letting first n ~ oc then e ~ 0 we obtain fn ~ f in LP(p) as n - - , co. It n ~ co. It remains to show that fn E ~+. By the C0-property of Pt, we have
'imllYi(P
t---~0
= l t~o i m i -[ n [J1/n t + l / n (P~f - P1/nf)ds - n__tfo t (P~f - f ) d s < t---~O-~ l i m n ( f l /n t+l/n IIP~f - Pl/nfll2ds + fot [IP~f - fll2ds) - O. Thus, fn e ~ ( L ) with Lfn - n ( P 1 / n f f ) e L I ( # ) N L ~ ( # ) , hence fn e ~+. (b) For any f c ~+(oz), let gn " - - f /~ n . One has gn "-+ f in L2(#) and o~(gn, gn) < o~(f, f). I f os176i s symmetric, by Proposition 0.1.7 (3) we have lim ~(gn,gn) - - ~ ( f , f ) -lim 8 ( f , gn) = lim 8(gn, f). Then gn ~ f
n--~c~
n---~oc
n--,c~
under ozl/2 as n ~ oc. For any n ~> 1, take any n ~> 1, take kn ~> 1 such that 8~/2(f
-
gk~,f - gk~) ~< -1. n
Next, by the -1 f0 ~ Psgknd s ~ gkn under r
Chapter 5
210
-
Log-Sobolev Inequalities and Semigroup Properties
1 / ~ Psgk~ ds ~ gk~ under 81/2 as ~ --+ 0. T h e n there exists l~ >~ 1 such t h a t
E
In
fn "-- In
Psgk~ds satisfies Sn ~ f under N1/2 as n -+ c~. Finally, by the J0
fn E ~+ for each n >~ 1. T h e n the proof is finished,
proof of (1) we have
r-I
Let p E eli0, (x)) with p ~ 2 for infinite # and p > 1 for finite #. Then for any f E ~+, (Psf) p(s) is differentiable in s under LI(#) with Lemma
5.1.2
d
ds (P~f Proof.
)p(~)
- p(s)(PsL f)
(p~)p(~)-I f
+
p,
(5.1.2)
(s)(P~f) p(~) log P~f .
Let s ~> 0 be fixed. For any ~ E ( - s , c~),
](Ps+~f)p(s+e)-(Psf)p(s) -p(s)(PsLf)(Psf)P(s)-l-p'(s)(Psf) p(s) log P~f] E
](Ps+ef)p(s) - (psf)p(s)
- p(s)(PsLf)(P~f) p(s)-I ]
+ I (P~+~f)p(~+~)r (P~+zf)P(~) - p'(s)(P~+~f) p(~) log P~+~fl + p'(s)l(Ps+ef) p(~) log P~+ef - (P~flP(~)log P~S[ --" /1 -~-/2 + / 3 .
Since for any a, b >~ 0 there exists d between a and b such t h a t
[ap - bp
-
pbp-l(a
-
b)l - pl(a
-
b)(d
p-1
-
bp-1)l <~ p[(a
-
b)(a
p-1
-
bp-1)[,
and since
lap_l _ bp_ll ~< { la - bl p-1 la - bl (p - 1)(a v b) p-2 there exists c =
I 11
c(s, Ilflloo)
if if
p-1~1, p-l>1,
> 0 such t h a t
- P LSI * I
I
- P fh
S _ PsLSlJrl ps+ S_ps s ips+.cS_Psflp(s)_l
if
p(s) >>.2,
if 1 < p ( s ) < 2 .
Thus, if p(s) ~> 2, by the fact t h a t
Ps+~f - Psf ~ PsLf in L2(#), we #(I1) e 0 as e --+ 0. If # is finite and p(s) > 1, since the L2-convergence is stronger t h a n the Ll-convergence, we still have #(I1) --+ 0 as e ~ 0. Similarly, we have #(/i) ~ 0 as e - , 0 for i = 2, 3. [3
5.1
Three boundedness properties of semigroups
211
Let (o~, ~(o~)) be a symmetric Dirichlet form and let f c ~+. If either p >~ 2 or p > 1 but # is finite and f >>.e for some e > O, then
L e m m a 5.1.3
# ( f p _ l L f ) <~ - 4 ( pp2 - 1_____E( __~) fp/2 , fp/2 )"
(5. 1.3)
Proof.
It is easy to see that under our condition one has fp/2 E ~ ( oz) (see Proposition 0.1.3) and fp-1 C L2(#). By the spectral representation,
8(fp/2 fp/2) _ lim }#((fp/2 _ ptfp/2)fp/2) t---*O
# ( f p - l L f) - t~olim~ # ( ( P t f - f)fp-1). So, it suffices to prove p2
#((fp/2 _ ptfp/2)fp/9.) << ~ # ( ( f 4(p - 1)
Ptf)fP-1),
t > 0.
(5.1.4)
For fixed t > 0, since # is a-finite and Pt is contractive in L l(#), there exists a unique symmetric measure Jt on ~ x ~ such that Jt(A x B) # ( 1 A P t l B ) , A , B E ~,~. Then, letting Pt "-- 1 - Ptl, we obtain
#((fP/2-ptfp/2)fp/2 ) -
(fP/2(x)- fP/2(y))fP/2(x)Jt(dx, d y ) +
fPptd#
>E
1/E
(fP/2(x) -- fP/2(y))aJt(dx, dy).
Similarly,
#(fP-l(f-Ptf))
-- #(PtfP)+2
>E (f ( x ) - f (y))(f ( x ) p - l - f (y)p-1)Jt(dx, dy).
Then (5.1.4) follows from the fact that
(ap/2 _ bp/2) 2 ~<
p2
(a - b)(a p-1 - bp-1),
4(p - 1)
a,b ~> 0, p > l .
T h e o r e m 5.1.4 Let (oz, ~(d~ be a symmetric Dirichlet form. (1) /f (5.1.1) holds then for any p >>.2 and q(t) "- 1 + ( p - 1)e 4t/C1 , 1
1
If # i~ ~nit~, th~n (5.1.1) i~pli~ (5.1.5) fo~ all p > I. (2) If (5.1.5) hold~ fo~ p- 2 and th~ abow q(t), th~n (5.1.I) hour.
D
(5.1.5)
212
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Proof. (a) Let us fix p so that p > 1 for finite # and p >~ 2 for infinite #. For any f E ~+ (with inf f > 0 if # is finite and p E (1, 2)), let h(t) := #((Ptf)q(t)),t >. O. It follows from (5.1.1), Lemmas 5.1.2 and 5.1.3 that h'(t) - q(t)#((LPtf)(Ptf) q(t)-l) + q'(t) q(t) #((Ptf)q(t) log(Pt f)q(t) ) ' (ptf)q(t)/2)
~< Cl.q'(t) - 4(q(t) - 1)oz((ptf)q(t)/2
q(t) q'(t) h(t) log h(t) + ~C2q'(t) h(t) + q(t) q(t) = q'(t) h(t)log h(t) + ~C2q'(t-------~)h(t). q(t) q(t) Since
Iq(t) it follows that h(t) -IIPtfl,q(t),
h'(t) - q(t)llPtfl[ q(t)-I d q(t) d---~llPtfllq(t) + q, (t)h(t) log IlPtfl[q(t). Then we arrive at
d
C2q'(t)
d---illPtfllq(t) < q(t) 2 IlPtfllq(t). Thus,
1
Finally, by Lemma 5.1.1, ~+ is dense in L~_(#). Moreover, if # is finite then for any f e 2 + one has f + e --, f in LP(#) as e ~ 0. Thus IlPtilp q(t) 1 (b) If (5.1.5) holds for p -
2, then for any f e ~+ with 1
#((Ptf) q(t)) <. exp [q(t)C2(2
#(f2) _ 1,
1
Since the equality holds at t = 0, by Lemma 5.1.2 we may take derivatives for both sides at t = 0 to obtain (5.1.1). Since g is symmetric, by Lemma 5.1.1, ~ is dense in ~ + ( 8 ) under 81/2. Hence (5.1.1) holds for all f E ~+(g~). Finally, for general f E ~ ( 8 ) , (5.1.1) follows from the fact that [fl E ~ + ( 8 ) and g(Ifl, Ifl) <~ 8 ( f , f). R e m a r k 5.1.1 Although Theorem 5.1.4 was proved only for symmetric Dirichlet forms, it holds also for some non-symmetric case. More precisely, Theorem 1.4.2 (1) holds as soon as (5.1.3) holds for all f E I..j Pts~/for some t~>0
5.1
Three boundedness properties of semigroups
213
set ~ dense in LP(#). As observed by Bakry (see e.g. [13]), it is in particular the case for a diffusion setting. For instance, let L := A § Z on a complete Riemannian manifold for some Cl-vector field Z such that the L-diffusion process has an invariant probability measure #. For any f E C ~ ( M ) , the class of all C~-functions on M which are constant outside some compact sets, so that for any p > 1 one has (P t f ) p E ~ ( L ) for all t > 0 and hence by chain rule,
0 -- # ( L ( P t f ) p) - p # ( ( P t f ) p - l L P t f ) +
4 ( p - l_______~)#(iV(ptf)p/212) .
P Thus, (5.1.3) holds for all f E [.j Ptsyr with s~' := C ~ ( M ) .
On the other
t~>0
hand, Theorem 5.1.4 (2) holds as soon as 1~+ is dense in @+(2) under o~11/2 for non-symmetric oz . This is often the case in applications. For instance, in the above diffusion case one has {f E C ~ ( M ) : f ~> 0} c @+ and is dense in 2 + ( 2 ) "-- { f e H2'l(#) " f >~ 0}. To apply Theorem 5.1.4 (2) to derive (5.1.1), one has to verify (5.1.5) for p = 2 and all t ~> 0. Indeed, by means of interpolation theorems, this can be realized by verifying the hyperboundedness of a single operator Pt.
Let (E, ~ ( 8 ) ) be a symmetric Dirichlet form. If there exist t > 0 and q > p > 1 such that [[Pt[[p~q < C < cc for some C > O, then (5.1.1) holds for C1 = t p q / ( q - p) and C2 = (pq log C ) / ( q - p). T h e o r e m 5.1.5
Proof.
Let r c [0, 1/2q). By Stein's interpolation theorem with P0 - 2, Pl = 2(1 - r)/(1 - 2r/p),p2 = p and q2 = q, we have q0 = 2 p q / ( p q - 2 r ( q - p ) ) and [[P~tl[2~qo < C ~. Thus, for any f c ~ + with # ( f 2 ) _ 1, we have
#((Prtf) q~ <~ C rq~
r e [0, 1/2q).
Since the equality holds at r = O, we may take derivatives for both sides at r = 0 and hence obtain from Lemma 5.1.2 that
2 t p ( f L f ) + 4(q - p) #(f2 log f) ~< 2 log C. Pq Thus, (5.1.1) holds for
C1 =
tpq q-p
and C2 =
pq log C and all f c ~+ (and q-p
hence all f E ~ ( 8 ) as explained in the proof of Theorem 5.1.4).
[-]
Next, we study the superboundedness and the ultraboundedness by using the following super log-Soblev inequality: #(f2 log f2) ~< r S ( f , f ) + 3(r), where ~ : (0, oc) ~ IR.
r > 0, f e 2 ( E ) , # ( f 2 ) _
1,
(5.1.6) ,
214
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Let (8, ~ ( 8 ) ) be a symmetric Dirichlet form. If (5.1.6) holds then Pt is superbounded with
T h e o r e m 5.1.6
IIPtl12-~4
[
~< exp L~f~ 1 (4t/log 3)]
t>0.
Conversely, if Pt is superbounded then (5.1.6) holds for f~(r) := 4log IlP~/4ll2-~4,
r > o.
Proof.
Let C 1 = r : = at~ log 3, by (5.1.6) one has (5.1.1) for C2 := ~(4t/log 3) and hence the first assertion follows from Theorem 4.1.4(1) with p - 2. The second assertion follows from Theorem 5.1.5 by taking p - 2, q - 4 and t - r/4 for given r > 0. [:]
Let (8, ~ ( E ) ) be a symmetric Dirichlet form. If Pt is ultrabounded then (5.1.6) holds for/3(r) "- 2 log IIP~/2112-~, r > o. Conv~rseZy, assume that (5.1.6) holds for some decreasing/3 E C(0, c~) (note that because 8 >>. O, we may always assume that fl is decreasing by replacing 13(r) with
T h e o r e m 5.1.7
inffl(s))
s>~r
'
if for some r E C[2, c r
that t "=
p f2 ~ ~(;)
dp < cr then
1 Proof. We need only to prove the second assertion since the first follows from Theorem 5.1.5. It is easy to see from Lemma 4.1.3 that (5.1.6) implies # ( f f log f) <~ - r # ( f f - l L f ) + ~ ( 4 ( p - 1 ) r / p ) p - l # ( f f ) + #(fP) log IIfllp (5.1.7) for all f E ~+, 2 < p < c~, r > 0. Let p and N be two function on [0, t) such that
p'(~)= ~op(~)' P(~)
;(0) = 2;
p'(s)/3(4(p(s) - 1)r o p(s)/p(s)) N'(s) =
p(s) 2
N(0) - 0 .
Since for f E ~+ one has d
d--~#((Psf)P(S))
d
d~llP~fllp(~) =
p(s)llPsf lIp(s)-1 "p(~)
p'(s) p(~) IIP~fllp(~)log [[P~fIip(~),
it follows from Lemma 5.1.2 and (5.1.7) that
d
d--~{e-
N(s)
IIP~fllp(~)}
P'(S) e-N(s) p(~)
)p(s)-I
P ((:))
p(s)llP~f]lp(~)_ 1 {#((Psi) p(~) log P~I) + p,
#((P~I
LP~f)
5.2
Spectral gap for hyperbounded operators N'(s)p(s)
p'(~)
215
#((p~f)p(~)) _ #((p~f)p(s)) log
IIP~fllp(~)} ~ o.
Therefore,
e-i(~)llP~f][p(~) ~ I]fl12,
f e ~ + , s e [O,t).
Then the proof is completed by noting that p(s) ~ c~ and hence N ( s ) 2~ ~(4(p - p2 1)r(p)/p) dp as s ~ t. Indeed, p(t) - oc follows from the fact that p(t) r(s) d s -
~ r(s)
p(~) [2
Finally, we connect (5.1.1) with the spectral gap.
T h e o r e m 5.1.8 Assume that # is a probability measure and (E, ~ ( 8 ) ) is a conservative Dirichlet form such that the spectral gap A1 > 0. If (5.1.1) holds for some C1, (?2 > 0, then
1
#(f2 log f2) < (C1 + -~1(62 -~- 2))E(f,
f ) + #(f2) log#(f2),
f e ~(8).
2
Conversely, if (5.1.1) holds for C2 - 0 then A1 ~ -'~-. ~1 Proof. By Deuschel-Stroock's inequality (see Proposition 6.1.1 below for a more general inequality) we have
Ent,(f2) .= #(f2 log f2) _ #(f2)log #(f2) < Ent,(f2) + 2#(f2), where f e L2(#) and f "- f inequality we obtain
#(f). Thus, by (5.1.1) and the Poincar6
': Entz(f 2) <~ C l ~ ( f , f ) + ( C 2 + 2 ) # ( f 2) <. ( C I - ~ -C2 ~ + 2~8(f f), A1
f e 2(~).
Hence we prove the first assertion. The second assertion follows from the more general result Corollary 5.2.2 (2) below. [2
5.2
Spectral gap for hyperbounded operators
Let (E, ~ , #) be a probability space, and P a symmetric contraction linear operator on L2(p) with P1 - 1. Then the spectral radius of P is R ( P ) - 1. We say that P has a spectral gap if RI(P) "-sup{[[Pf[[L2(~)" # ( f 2 ) _ 1 , # ( f ) -
0} < 1.
Chapter 5
216
Log-Sobolev Inequalities and Semigroup Properties
Recall that P is called hyperbounded if 5 ( P ) "-IIPII~(,)_~L~(,) < c~.
We intend to find reasonable condition for the hyperbounded operator to possess spectral gap. It turns out that the answer is NO for non-ergodic positive definite operators. Recall that P is called positive definite if # ( f P f ) >~ 0 for all f E L2(#), and is called ergodic if n
lim -1 ~-~ p k f _ # ( f ) , n---~c~ n
f C L2(#),
k=l
where and in the sequel the limit is taken in L2(#). Let ~ := { f " P f - f } , and let 7r" L2(#) --, 3if be the orthogonal projection. For positive definite P it follows from the spectral representation that lim p n f _ 7rf + lim n---~c~
n---~c~
fo1- )~ndE~(f) -
7rf,
f E L2(#),
where {E~ : /~ E [0, 1]} is the resolution of the identity of P. Therefore, it is easy to see that the ergodicity is equivalent to each of the following two statements: (a) For any f e L2(#), p n f __~ # ( f ) as n ~ c~; (b) For any f E L2(#), /f P f - f then f is constant. Obviously, (a) (and hence, the ergodicity) implies the positivity improving property: (c) For any A , B with # ( A ) # ( B ) > O, there exists n ~ 1 such that # ( 1 A p n l B ) > O. Conversely, if P is positivity-preserving, i.e. P f >~ 0 for all f ~> 0, then (c) implies (b) and hence is equivalent to the ergodicity. Indeed, for P f - f with # ( f ) - 0 , one has P f + ~ ( P f ) + - f+, where f + "- f V 0. But # ( P f + ) #(f+), then P f+ - f+. Thus, for any n/> 1 and any ~ > 0, 0-
#(f+ f-) = #(f-pnf+)
>~ ~2#(l{f_>e}pnl{f+>e})"
By (c) one has # ( f - > ~)#(f+ > ~) - 0 for all ~ > 0. Hence, either # ( f + ) = 0 or # ( f - ) - 0. But # ( f ) = 0 implies # ( f - ) = # ( f + ) , one concludes that f - 0. Therefore, (c)implies (b). The following result says that the hyperbound estimate 5(P) < 2 is the minimal sufficient condition for the existence of spectral gap.
5.2
Spectral gap for hyperbounded operators
Theorem
217
(1) /f 5(P) < 2 then P has a spectral gap with
5.2.1
} 1/2
V/25(p )
RI(P) <
3V/25(p) _ 4V/5(p ) _ 1
< 1.
(5.2.1)
(2) There exists a Markov operator P with 5(P) = 2 and R i ( P ) = 1 (i.e., P does not have spectral gap). (3) If P is positive definite and 5 "- 5(P) < 2, then [[P~[[L2(~)_~L4(~) -- 1 for all r >~ ro, where log {3 + 3(175 - 8 - 12V/25(5 - 1 ) ) / 1 6 } I' 0 : ~
< oc.
log (3 - 4v/5 - 1 / x / ~ )
Proof. (a) For e E (0,1) and f E f2(#) w i t h # ( f 2 ) 1 a n d # ( f ) - 0 , let g "- V~ + x / 1 - r We assume that # ( ( p f ) 3 ) ~> 0, otherwise simply use - f to replace f. Since p ( P f ) - p ( f P l ) - p ( f ) - 0 and since # ( ( p f)4) /> p ( ( p f ) 2 ) 2 it follows that 5(P) "-[[P[[~2 (#)__,L4(#) >/#((Pg) 4) = r + (1 - r
+ 6r
- r
+ 4~3/2 v/1 - e # ( P f ) + 4v~(1 - ~)3/2#((pf)3) /> (1 - ~)2#((p f)2)2 + 6e(1 - c)#((P f ) 2) + ~2. Thus,
# ( ( p f ) 2 ) < V/8r + 5 ( P ) - 3~ =. h(r
#(f)-
I-r
0, # ( f 2 ) _
1, r e (0, 1).
Therefore,
RI (P)2 <<~(6(P)) "-
inf h(r
~C(0,1)
(5.2.2)
It is easy to see that when 5(P) = 1 one has ~(1)-
v/8c 2 + 1 - 3r 1 --. ~-~1 1-r 3 lira
Next, for 1 < 5(P) < 2, the minimum of h(r over ~ E (0, 1) is reached at c(5(P)) "- 5 ( P ) Indeed, one has (where "r h'(s)-0r162 r
~3 V/25(p)(5(p) _ 1)
means "is equivalent to") V/8r 2 + 5 ( P ) - 3
8r + 5(P) - 3V/8r 2 + 5(P)
+v/8c 2+5(P)-3~-0
218
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
r162Sa2 - 166(P)~ + 96(P) - 6(P) 2 = 0
r162~ - 6 ( P ) -
~i
V/166(P) 2 _ 2(96(P)
-
6(P) 2) -" e(6(P))
Thus, ~(6(P)) = h(s(6(P))). It remains to calculate h(s(6)) for 6 /> 1. Observing that
8 ( 6 - ~ 3 V/26(6
-
1)) 2 + 6 = 8 6 2 + 9 6 ( 6 - I ) - 1 2 6 V / 2 6 ( 6 - I ) + 5
= 86(6 - 1 ) - 126V/26(6- 1 ) + 962 = ( 3 6 - 2V/26(6- 1))2, and noting that 35 > 2V/26(6- i) for 6 ~> 1, one has
~(6) = h(r
V/26(6- I) --- 3%//26(6 _ 1)- 4(6- I)
3v/~-
4v/6
-
1
Therefore, (5.2.1)follows from (5.2.2). (b) Let E -
1 {0, 1},#(0) = #(1) = ~ and P = I (the identity operator).
Then P does not have any spectral gap. Next, let fl(0) - - 1 and f1(1) = 1. For any f 6 5 2 (#) with #(f2) = 1, there exists Cl, C2 E ]~ such that c21+ c2 = 1 and f = Cl + c2fl. Then # ( ( p f ) 4 ) = #(f4) = Cla + c2a + 6ClC2 =
+
c2)2 + 4c2c~
~< sup { l + 4 r ( 1 - r ) } = 2 rE(O,1)
1 with equality holds when c21 = c2 - ~. Thus, 6(P) = 2. (c) For f e L2(#) with #(f2) = 1, let / "= f - #(f). Let R denote the upper bound of RI(P) given in (1), and simply denote 6 = 6(P). One has R < 1 as 6 < 2. Since P is positive definite, P~ is well-defined for r/> 1 and we have # ( p r ] ) = # ( / ) = 0. Hence # ( ( p r f ) 4 ) _ #(f)4 + 4 # ( f ) # ( ( p r / ) 3 ) + 6#(f)2#((prf)2) + #((pr/)4). Noting that
4#(f)#((Pr]) 3) <. 4[#(f)[~/#((P~/)a)#((Pr/) 2) <2t#(f)2#((Pr]) 2) + 2t-l#((Pr])4), t > O, #((pr])2) < R2~#(]2), #((pr])4) < 6#((pr-1])2)2 < 6R4(r-1)#(]2)2,
5.2
Spectral gap for hyperbounded operators
219
we obtain
#((pr f)4) <
t>0. (5.2.3)
Letting
t--~l{V/9+25R2r-4 + 52R 4r-8 + 5R 2r-4 - 3 } , we have R 2r (3
+ t) - 5R 4(r-1) (1 + 2/t) = 1 R 2 r { 3 + 5R 2r-4 + V/9 + 25R 2r-4 + 52R4r-8} 2
It suffices to prove that (5.2.3) that
-" h(r).
h(ro) ~< 1. Indeed, if h(ro) ~< 1 then it follows from
#((pro f)4) <~{#(f)2 + #(f2)}2 _ 1 for all f e L2(#) with # ( f 2 ) _ Let
1. log(3 + s) and log R -2
s - 35//(16R 4), one has r0 5
3+2s1 Moreover, R 2~~ = ~ . 3+s
h(ro)
(3 +
s)R 4
35 >/3 -~ 8R 4
5 3R 4 > 0 .
Then
5R_4 ~ ( 9 + 25R-4 +
1 3~ ~< 1 w 2------~ 6+ 3+s
3+s
)9+
r
r
3+s 3+s 65R-4(1 + s) 4s 2 + 12s ~> 3+s 12s 2 + 2s 3 + 18s l+s
+
(3 + s) 2
52R-8 (3 + s) 2
>/35R -4.
Since s 2 ~> 2 s - 1, we have 12s 2 + 2s 3 + 18s >~ 12s 2 + 2 s ( 2 s - 1 ) + 18s - 16s(1 + s). Therefore,
h(ro) <~ 1 follows from the fact that 16s(1 + s) = 1 6 s - 35R -4. l+s
220
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
C o r o l l a r y 5.2.2 Let (Pt)t~o be the symmetric Co-contraction semigroup generated by a negative definite self-adjoint operator (L, ~(L)) on L2(#) with L1 = O. Let g a p ( L ) : = i n f { - # ( f L f ) " # (f ) - O , #(f2) _ 1} be the spectral gap
ofL (1) If there exists t > 0 such that 5(Pt) < 2, then gap(L) ) - ~
1
log
V/25(Pt) aV/25(Pt) - 4V/5(Pt) - 1
> O.
(5.2.4)
(2) If Pt is Markovian and there holds the defective log-Sobolev inequality:
(5.2.5)
f e ~(L), # ( f 2 ) = 1
#(f2 log f2) ~ - C l # ( f L f ) + Ca, for some C1 > 0 and C2 E [0, log 2), then
2 v/2e C2 gap(L) ~> - ~ log > 0. C1 log 3 3x/2e C2 - 4x/e C2 - 1
(5.2.6)
2
In particular, if (5.2.5) holds for C2 = 0 then gap(L) ~> ~11"
(3) There exists an example such that (5.2.5) holds for C1 - 0 log 2 but gap(L) - 0. Proof.
and C2 -
(a) If 5(Pt) < 2 for some t > 0, then by Theorem 5.2.1 one has V/25(Pt)
#((Ptf) 2) <. av/2
(p
=: st < 1,
) -
-
#(f) -- O, #(f2) _ 1.
i
Thus, it follows from Lemma 1.4.8 that #((Psf) 2) ~ #((Ptf)2)s/t#(f2) 1-sIt ~ ~t It,
s e [0, t], # ( f )
-- O, # ( f 2 )
_ 1.
Since all three terms in the above formula are equal when s = 0, it follows that d# 2#(f L f) - ds
1
((P~f)2)l s=0 ~< ~- log et,
f E @(L), # ( f ) -
Thus, we arrive at the Poincar~ inequality #(f2) _ #(f)2 <
and hence (5.2.4) holds.
2t # ( f L f), log ct
f E ~(L),
0, # ( f 2 ) _ 1.
5.2
Spectral gap for hyperbounded operators
221
(b) If (5.2.5) holds then by Theorem 5.1.4 one has IlPt[lL2(~)_,id(~) <. e C~/4 C1 for t : = 4 log 3. Hence (5.2.6) follows from (5.2.4). Finally, let E = {0, 1} 1 with #(0) = #(1) - ~. Let L - 0 (hence the corresponding form s - 0) which does not have spectral gap. We have, for any function f with #(f2) = 1, 1
p(f2 log f2) _ ~{f(0)2 log f(0) 2 + f(1) 2 log f(1) 2 } 1
sup
log
+ (2 -
)log(2 -
Therefore, (5.2.5) holds for C1 --0 and C2 = log 2.
= log 2.
V1
Corollary 5.2.2 shows that C2 < log 2 is the optimal sufficient condition for the defective log-Sobolev inequality (5.2.5) to imply gap(L) > 0. Below we present the optimal sufficient condition for the defective Poinca% inequality to imply the existence of spectral gap. T h e o r e m 5.2.3 Let (L, ~(L)) be the generator associated with a conservative Dirichlet form (8, ~ ( 8 ) ) . If there exist C1 > 0 and 6"2 E [1,2) such that #(f2) < C18(f, f ) + C2#([f[) 2,
f e ~(s
(5.2.7)
then
gap(L) >~
1 - v/C2(C2- 1)/2 > 0. 2C1
On the other hand, there exits a conservative Dirichlet form such that (5.2.7) holds for C2 = 2 but gap(L) = 0. To prove Theorem 5.2.3 we need the following lemma.
L e m m a 5.2.4 Let (L, ~ ( L ) ) be associated with a Dirichlet form (~, ~ ( 8 ) ) . Assume that there are four constants al, c~2,/31,/~2 > 0 such that p(f2) <
f e ~(L), # ( f ) = 0, (5.2.8)
and
#(f2) ~< a28(f, f ) + / 3 2 # ( I f I) 2
f e 2(e).
If ~1 ~2 < 2, then gap(L) ~> 1 -- 4/~1/~2/2 ~ O.
2(~1 v c~2)
(5.2.9)
+
"~
i.-i~
e-i.-
0
l'o
+
~
~
~%
~
+
~
"~
oq
"
,..,.
0
go
CO 84
E~ ~
~
~: ~
o
e,1,-
c,~
ol
I
~
~. <~
l:: ~
o
o'q
~
" ~\V
~:~1 ~'-~
II
0q
~
.-.
F1
9
i-.i.
,-,-
o
o
i-t
~D
('1)
i-.1~
~D
~.~.
~
~-"
"l::
~
m ~
~
~
9
o
B',~
~
"
~
~
/A
I
D
I
+
+
+
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~
I
/A
'~
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I
II ~
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-
,,~
I
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B
~
II
i,o+ ~o "~-.
~
~
~D
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o~ ~
~~
II
I ~+
~ ~
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I
;..,
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~
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I "~
V
~~
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9
"~ 9
9
9
"~ ~
~'~ > ~
II
~ ~"
("1)
0
~
-J
II ~
~
~
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,~
9
~=~~
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I
~-" ~
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g
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~
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~\v
II ~'~ ~ o >.~~ ~ "II
~ ~
=
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~
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~'~=
01~
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o ,a
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d)
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I---I
o
~-
o
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~D
5.2
Spectral gap for hyperbounded operators
223
Similarly, the same holds for (sup f)2 in place of (inf f)2. Thus, p(f2) ~< C18(f, f) + (C2 - 1){ (sup f)2 A (inf f)2},
f e @(3~), # ( f ) -- 0. (5.2.13)
Then the first assertion follows from Lemma 5.2.4. 1 On the other hand, let E - {0, 1} with #(0) = #(1) - ~. Let L - 0 (hence the corresponding form oz = 0) which does not have spectral gap. We have, for any function f, 1 )2 1 p(f2) _ ~{f(O)2 + f ( l } ~< ~(If(O)l +
[f(1)l )2 - 2~(Ifl )2 9
Hence (5.2.7) holds for C1 = 0 and (72 = 2. K] Finally, we intend to construct an example to show that, even in the ergodic case, the hyperboundedness is insufficient for the existence of spectral gap. Consider, for instance, E0 := {0, 1} and 1
Lof(i) "- ~[f(j) - f(i)], Let #0(0)
#0(1) -
i,j E Eo, j r i.
1
~. Then L0 is self-adjoint in L2(#0) with spectrum
{0, - 1}, where the non-trivial eigenfunction is given by u(0) = - 1, u(1) = 1. Moreover, the corresponding Markov semigroup is determined by
p(O) f ._ #(f) + e_t#(fu)u, N o w , let E
-
ENo "-- { ( X n ) n )
t >1 O, f e n 2 (#o). 1}
1 " x n e E o , Tt )
and let # - #~ be the
product measure. Consider the semigroup
- H p,(o n~>l where Pt(~) "= Pt(~ acting on the n-th space. Therefore, Pt is a symmetric Co oo
Markov semigroup on L2(#) generated by L "- ~
L~n) with domain ~ ( L )
n--1 oo
L~n)f converges in L2(#),
containing all functions f c L2(#) such that ~ n=l
where L~n) stands for the operator L0 acting on the n-th variable. It is clear that the spectrum of L contains merely eigenvalues { 0 , - 1 , - 2 , . . . }, and the multiplicity of each eigenvalue is infinite. Let ]H[i denote the eigenspace with respect to the eigenvalue - i . One has 1HI0= IR and
]}'][1 - - s p a n { f n ' f n ( X ) " - - U ( X n ) , X
e E , n >~ 1}.
224
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Let ~i " L2(# N) --* Hi be the orthogonal projection, i ~> 1. Then the (X)
spectral representation implies L f - - ~
iTri(f), f e ~ ( L ) . We then define i--1
a new self-adjoint operator on L2(#) via the spectral representation
-
n1(/fn)/n' i~l
f e ~(L),
(5.2.14)
n--1
where ~(],) contains all f e L2(#) such that the right-hand side of (5.2.14) makes sense in L2(#). Let Pt be the corresponding C0-semigroup. It is clear that P t f - Ptf, f e H~I , (5.2.15) where IHI~ denotes the orthogonal complement of H1.
Proposition 5.2.5
Pt is a hyperbounded, ergodic, symmetric Co-contraction semigroup without spectral gap.
1 --fn, n >~ 1, there is no spectral gap. Next, let n f E L2(#) such that P t f - f for some t > 0, and let f" be the orthogonal Proof.
Since L f n -
(X)
projection of f onto ]H[~. We have ~--~(1- e - t / n ) # ( f f n ) f n
-- 0 and hence
n--1
i t ( f l u ) = 0 for all n ~> 1. Thus, f - f" and P t f = P t f - f . By the ergodicity of Pt the function f has to be constant. Therefore,/St is ergodic. It remains to prove the hyperboundedness of Pt. For f e L 2 ( # ) w i t h #(f2) ~< 1, one has CX3 f -- f ' w f u - - ~ - ~ C n f n W f U n=l (X)
for some
sequence
{Cn }
C ]~
with ~
Cn2 -- #(f,2) ~ 1, where f' and f" denote,
n--0
respectively, the orthogonal projections of f onto IH[1 and H1-k. Since # is the product measure and since #(f~) - # ( f a) _ 0 and f~ - fn4 - 1 for all n ~> 1, it follows from Fatou's lemma that N
4
d# .((J~ ~ lirai. (~-~cne-'In'n) N--*c~ (X)
4 <~ ~-~Cn_~_ n--1
n--1 (X)
~-~ amen 2 2" < m,n=l
2#(f'2) 2
(5.2.16)
5.3
Concentration of measures for log-Sobolev inequalities
225
On the other hand, since p(0) is hypercontractive, so is Pt (see e.g. [99]). Hence there exists t > 0 (independent of f) such that #((/St f,,)4 ) _ #((ptff,)4) ~ #(fH2)2. Combining this with (5.2.16) we arrive at #((ptf)4) ~< 23(#((/5tf,)4 ) + #((ptf,,)4)} ~< 24[#(f,2) + #(f,,2)]2 -- 24#(f2) 2. Therefore,/St is hyperbounded.
Concentration inequalities
5.3
of m e a s u r e s
for l o g - S o b o l e v
In this section we assume that # is a probability measure. We intend to study the concentration of # using (5.1.1) and (5.1.6). T h e o r e m 5.3.1 Let p E ~ ( L s ) with L3(p) ~ 1. We have c := log #(ep) E I~. Let a = 1 for local E and = 2 otherwise. (1) If (5.1.1) holds then #(e )'p) ~< exp[cA + C l a A ( A - 1 ) + C 2 ( A - 1)],
/> 1.
(5.3.1)
(2) If (5.1.6) holds, then for any strictly positive r E C[0, oc), p(e ~p) ~ e x p
E f cA+A
{ar(s) + ~(r(S))s 2 }ds] ,
A ~> 1.
(5.3.2)
Proof. Since (5.1.1) implies (3.3.3) for/3(r) - - e - ( ~ ( l + r - 1 ) for some 5 > 0, by Theorem 3.3.19 we have p(e ~p) < c~ for any A > 0. In particular, c "- log #(ep) is finite. By first consider (p V ( - n ) ) A n then let n + c~, we may assume that Next, let h(A)"- p(e~P), A > 0. It follows from (5.1.1) that
1 h'(A) - #(pe ~p) - ~#(e~P log e ~p) ~ ~C1 8(e~p/2 , e~p/2)+ ~C2 h(A)+h(A)log h(A) By Lemma 1.2.4 and Remark 1.2.1 we arrive at
h' (A) <~ aCI A h(A) + -~-C2h(A) + ~h(A)log h(A) 4
Thus, d
log h(A)
h'(A) }- Ah(A)
log h(A) aC1 C2 A2 ~ - ~ + --7" A
226
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
This implies (5.3.1). Similarly, if (5.1.6) holds then for any strictly positive r E C[0, c~) we have d log h(A) ar(s) , d-A{ A }--4 This proves (5.3.2).
)Q
'
C o r o l l a r y 5.3.2 Let Ls(p) ~ 1. If (5.1.1) holds then #(e ~p2) < c~ for any A < 1/(4Cla). Consequently, if (5.1.6) holds then #(e ~p2) < c~ for all A > O.
Proof.
For any s > 0, let ~(dA)"= e-(Cla+~)~2dA. By (5.3.1) we have
oo >
d#
-
e)'Pu(dA)
/ C l a -+-'-C)~- 2v/C1fla -~-~ )2] dA - {[ V
~ exp [p2/4(Cla + e)]d# / l O Oexp [
(
):j:
/> # exp [p2/4(Cla + ~)] v/C1 a + ~
C~a+ee
-,, dA.
This implies the first assertion since s > 0 is arbitrary, and hence the second assertion follows by the arbitrariness of r > 0 in (5.1.6). By Corollary 5.3.2, (5.1.6) implies a concentration stronger than the exponential integrability of p2. So, there should be a function 9 determined by g such that ~(r)r -2 --, c~ as r --, c~ and #(e ~(pl)) < c~. Since for p E ~ ( L s ) one has IpI E ~ ( L s ) and Ls(iPI) ~ is(p), we may assume that p ~> 0.
Let p E ~ ( L s ) be nonuegative with LE(p) <~ 1. Given strictly increasing function r E C 1([0, c~)) with r - 0, let
L e m m a 5.3.3
qh(A) . -
/o
r
~(A)
If ~ := sup ~" < c ~ f o r s o m e R > ~ [n,~) one has #(exp[+(p)]l{p>~ro}) ~ Proof.
Let u(s, A ) " - s s
.__
/o
~)-1
(r)dr, A > 0.
1, then for any ro > 0 with r
--
# (e)'P- ~()')) dA.
(5.3.3)
0 and hence
~P(A)- ~(s). one has d u ( s , r
u(s, r = u(0, r - 0. Next, O~u(s, A)[~=r - s- r or 02u(s, A) - - ~"(A) >~ - ~ for A ~> R. Then u(s, A) >~ -a(A - r /> r and s >/r0. Thus, or eU(S'X)dA ~>
e_,(x_O(s)):/:dA _ (~)
~> R,
.
- 0 and for
5.3
Concentration of measures for log-Sobolev inequalities
227
Therefore, e~(S) ~ (2-~) 1/2 flC~ e s~- e(~)dA and hence (5.3.3) follows. Theorem
5.3.4
A s s u m e that (5.1.6) holds with some decreasing and con-
~(~)
tinuous f u n c t i o n / 3 , and let c, a and p be in (5.3.2). Let H ( r ) " -
~/(r) " -
c +2al H - l ( a r 2)
L~t
~(~).-
V]
if r E [0, 1], otherwise,
r
and r
"- 2a fo r 7(s)ds,
~-~(~)d~, ~ h ~ ~ - l ( a ) . _ ~ fo~ ~ >1 r
s > 0.
Th~n Io~ any
p >1 0 with L s ( p ) <<. 1, one has ~(exp[~(p)]) < ec and hence IIPlI~ ~ r
Without loss of generality we assume that ~ is strictly positive and smooth. Otherwise, we may take a sequence {~n} with such properties and ~n >/3, ~ ~< 0, ~ ~ / 3 uniformly, then repeat the proof below for ~n in place of/3 and finally obtain the desired result by letting n ~ ec. Assume that c > 0, otherwise we have p - 0, #-a.e. Let O(A) "= Pro@
f0 a ~(r)dr. It is easy to see that
O(A) - 2aA
/0
a")/(r)
- / 3 ( " / ( r ) ) / r 2 for
",/(r)dr - (c + 1)A + 2aA
/1
r
> 1 and hence
3'(r)dr,
A > 0.
Thus, by (5.3.1), #(exp[Ap]) <~ exp [cA +2aA
/1
7(r)dr
1
- e x p [ - A + ~P(A)],
A > 0. (5.3.4)
Since ~/is decreasing on [1, oc), we have 0 ~ ~P"(r) ~ 4a~/(1) for r/> 1. If r - oc then r "- r is well-defined on (0, oc) and r o r s, r = 0, r = oc. Thus, the first assertion follows from Lemma 5.3.3 and (5.3.4). For the case that ~(oc) < oc, let ~ ( r ) - ~ ( r ) + e r and ~ ( A ) -
/0
r
(r)dr,
e > 0. Since ~-1 ~ r as e ~ 0 and r162 - ec, we have ~ - I ( R ) > R for some fixed R E (1, ~(ec)) and all small enough e > 0 (recall that r > ~(1) > 1). By applying (5.a.a) and (5.3.4) to ~e and Oe, we obtain (x)
exp[-A]dA, and hence #(exp[~(p)]) < oc by letting r ~ 0.
Z]
228
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
We notice that the choice of 7 in Theorem 5.3.4 is optimal in terms of (5.3.2) and (5.3.3) in the sense that for any positive 9 e C[0, oc),
a~/(r) +/3(~(r))r 2 >~ ~7(r)a + 13(7(r))2r 2 = aT(r),
r > 1,
since aT(r) = l~(7(r))/r 2 for r > 1 and 3 is decreasing. Theorem
5.3.5
In the situation of Theorem 5.3.4. Define
~(s)=inf
{
/1
A>l'2a
)
H - l ( a r 2 ) d r >>.s ,
s > O,
where H - l ( r ) "- sup{A > 0" H(A) ) r} and we set inf Z - oo by convention. Then for any p ) 0 with Ls(p) <. 1, #(exp[(hp - c - a)~(ep)]) < oc,
a > 0, e e (0, 1), 5 e (0, 1 - e ) .
(5.3.5)
Consequently, Ilpll~ < o~ if f F H-l(ar2) dr < c~. Proof.
Let
h(A) "- exp [ - (c + a)A - 2aA /1 H-l(ar2)dr]
A>O.
By (5.3.4) with 7 ( r ) " - - H - l ( a r 2) we have
1 /, d#
exp[Ap]h(A)dA-
For any r E (0, 1) and 5 E (0, 1 - r exp[Ap]h(A)dA ~>
il (:X)
#(exp[Ap])h(A)dA < oc.
(5.3.6)
there exists C(e, 5) > 0 such that exp [ ( ( 1 - a)p - c - a)A]dA
exp[(hp - c - a)~(r
- C(r 5).
V1
Therefore, (5.3.6)implies (5.3.5).
C o r o l l a r y 5 . 3 . 6 Assume that (5.1.6) holds for t3(r) - )tl -}-/~2 r-5(/~l, )t2, (~ > 0) Let p >~ 0 be such that Ls(p) <. 1. Then there exists a = a(5) > 0 such that #(exp[ap25/(5-1)]) < c~ if 5 > 1, #(exp[exp[ap]]) < oc
if 5 -- 1,
Ilpllor < oo
/i 5
< 1.
(5.3.7)
5.4
229
Logarithmic Sobolev inequalities for jump processes
Proof.
We use the notation in Theorems 5.3.4 and 5.3.5. Take r0 > 0 such
that H(ro) - 3(r0) < 1. Let Cl > A2 be such that fl(r) < clr -~ for r E (0, r0]. r0
We have r ~< r0 if H(r) >~ 1 and hence 7(A) "-- H - I ( a A 2) - s u p { r / > O ' H ( r ) ~ aA 2} ~< sup{r >/0" clr -1-5 ~/aA 2} = [Cl/(aA2)] 1/(1+5)
A>I.
By Theorem 5.3.5 we have Ilpll < ~ if 6 < 1. Next, it is easy to see that there exists c2 - c2(6) > 0 such that
{ C2)~(6+1)/(5-1) ~(A) >~ exp[c2A],
6>1, 6=1.
Then the proof is completed by Theorem 5.3.5.
Logarithmic Sobolev inequalities for j u m p processes
5.4
Let (E, ~ , #) be a measure space and (E, ~ ( 8 ) ) the quadric form introduced in the beginning of w We first study (5.1.1) and (5.1.6) for this form using isoperimetric constants, then provide criteria for these two inequalities in the context of birth-death processes.
5.4.1
Isoperimetric inequalities
Following the line of w
let ~ and S satisfy (1.3.7) and define
J(~) := 1{~>~ J, K (a) "= l{s>~ K, ~ S~
a E (0, 1].
For given r c (0, oc), let j(1/2) (A • A c) + K (1/2) (A) k(r) "-
inf ~(A)~(0,H
#(A) v/log(1 + p(A) -1)
where we set info - oc by convention. T h e o r e m 5.4.1 (1) If there exists r > 0 such that k(r) > 0 then (5.1.1) holds for C1 "- 9/k(r) 2 and C2 "- 2 + 2log + r -1 + 2 r - l ( 1 + log + r - l ) . (2) /f k(0) "- lim k(r) - c~, then (5.1.6) holds with
r---.O
~(r) "- inf {2 + 2log + s -1 + 2s-1(1 + l o g + s - 1 ) ' s
> 0,9 <~ rk2(s)}.
I
r.-,
~
~.,
ID-,
I ~
+
~
.~
9
~1~
~
+
~
I
t~
+
m
:~ ~
~>
0
o
.
"~
~o
~
~
~
~X
t~l~.-'
~
u
Ca
~,-,-
"
~
~
0
~-~
~
\V
I
E'
I.-,.
.
-~ I
I
~1~
I
+ ..~
O0q
+
~
+
~
~
.
"-,-'
~
o
~
t~
.
+
~
"l:
J~--.~
I
o~
+
I
'-" 0
.
~
~
+
L ~
8L
~
~-
~
89
0
+
I
I
89
+
~"
8
~
~
~
+
~
~
~
~
~i~
~
~.
J
I
.-y,
~
~
~::-"
~
\V
-.
\V
B iJo
b~
5.4
Logarithmic Sobolev inequalities for jump processes
231
it follows that I ~ ~3J;A•
[ f ( x ) - f(y)[ { ( [ f ( x ) l l l + log + f(x) 2)
dy )
V (If(Y)IV/1 + log + f(y)2)}'~(1/2)(dx,
<~3{~(f'f) L [f(x)2(l+log+f2(x)) 2 zxxE~x
dy)}
+ f(y)2(1 + log +
~< 3V/e(f, S)#(f2(1 + log+ f2)), where the last step is due to (1.3.7). Combining this with (5.4.1) we arrive at
#(f2 log+ f2) ~
-t- 7"-1
(1 + log + r - 1 )
1
q- ~ -~-
9
2k(r) 2
8(f, f)
1 + ~#(f2(1 +log + f2 )).
This implies that
#(f2 log+ f2) ~< 5.4.2
98(f, f) + 2 + 2 log + r - 1 + 2r-1(1 + log + r-l).
[-1
C r i t e r i a for b i r t h - d e a t h p r o c e s s e s
Due to an observation of Bobkov and G6tze [22], (5.1.1) is equivalent to the Poinca% inequality with an Orlicz norm in place of the L2-norm. This enables us to study (5.1.1) by using Hardy type inequality. Let ~ be a class of nonnegative functions on E := Z+. We define the N-norm by Ilfl[~ := sup #([fig)" gE~
For
En := {0,'-" , n}, n ) 0, consider the Poincar6 type inequality: IIf21l~ <~A~8(f , f),
flEn =0,
(5.4.2)
where 8 is given by (1.3.5) and A~ is the smallest positive constant such that (5.4.2) holds. Let i
Bn "- sup ~-'~(#ibj)-l [[1Eg[l~. i ) n j=0
Then the discrete Hardy inequality implies the following criteria for (5.4.2).
232
Chapter 5
Proposition 5.4.2
Log-Sobolev Inequalities and Semigroup Properties
Let (8, ~ ( 8 ) ) be given by (1.3.5). One :has Bn <. An <.
4Bn for all n >~ O. Proof. For given g E ~ , let 13i " = # i + n b i + n , ozi " - g i + n + l # i + n + l , i >f O. Given f with flE~ = 0, let h(i) "= f ( i + l + n) - f ( i + n), i ~ O. Then i
h(j) = f (i + n + 1). Hence Lemma 1.3.10 implies j=o
sup( i~n
i
" 9_
#jbj
(~
sup
gj+l#j+l <~ j~i
)
gi+l#i+lf(i "+-1) 2
~ ( f , f ) = l , f En=O i~n i
.< 4su i~n
"= # j b j
~
gj+l#j+l.
j >>.i
[i]
Taking sup over g E ~ we complete the proof. To apply Proposition 5.4.2 to (5.1.1) and (5.1.6), let
sup(
i
"=
1
O0
~ O0,
n>.O.
j=i+l
j=~+l
T h e o r e m 5.4.3
Let (8, ~ ( 8 ) ) be given by (1.3.5). (1) (5.1.1) holds for some C1, C2 > 0 (hence for some C1 > 0 and C2 - O) if and only if 50 < c~. (2) (5.1.6) holds for some ~ if and only if 5n ---* 0 as n ---, c~.
Proof. (a) Let ~ be a N-function, i.e. a nonnegative, continuous, convex and even function satisfying ~(x) - 0 if and only if x - 0, lim ~ ( x ) / x - 0, x---*0
lim ~ (x)/x = c~. Let X----~ OO
e*(y) := ~up(xlyl- e(z) z/> 0),
yE~.
Then ~* is once again a N-function. Let ~ " - {g ~> 0 9 #(~*(g)) 1}, IIflle " - i n f { A > 0 " # ( ~ ( f / A ) ) <~ 1}. By [154, Proposition 3.3.4] we have (5.4.3) Ilfll ~ < I]fJJ~ ~ 2JJfJl ~, Next, by the definition of I1" II~, one has
1 PIl[~,oo)[[~-- ~_l(~([x 'oo))_1),
x i> 0.
5.4
Logarithmic Sobolev inequalities for jump processes
233
Now, let us take ~P(x)"-Ixl log(1 + Ixl). Then it is easy to see that (see [138, Lemma 2.5]) t ~ ~ - l ( t ) ~ at c log t log t 1 for some c > 0 and all t >f ~ . 1
-
Thus, there exists c > 0 such that for this #0
one has
1Bn < an <~ cBn,
n >~O .
c Moreover, for this ~ one has
(5.4.4)
#(f2 log f2) _ c ~< IIf211 ~ <~ p(f2 log f2) + c
(5.4.5)
for some constant c > 0 and all f with #(f2) _ 1.
(b) If
<
th n (5.a.3), (5.a.4), (5.4.5) and Proposition 5.4.2 imply #(f2) _ 1, flE~ - O.
#(f2 log f2) ~< 4C(~nE(f, f) + c, Then for any f with # ( f 2 ) _ 1, n
#(f2 log f2) ~< )-~(f(i) 2 log y(i)2)#i + 4ChnS(flE,~, flE,~) + C i=0 n
~< ~-~(f (i) 2 log f (i)2)#i i=0
+ 4C5n {s
f) + #nbn(f(n + 1)
Since f(i) 2 ~< #~-1 for all i ~> 0, there exists
O/n > O
p(f2 log f2) ~ 4ChnS(f, f) + oLn
-
-
f(n)) 2 } + c.
such that #(f2) = 1.
,
Thus, 5o < c~ implies (5.1.1) while (~n ---+ 0 a ~ n ~ (x) implies (5.1.6) for some ~. Moreover, if 5o < c~ one has 5o < c~ and hence according to Theorem 1.3.11 the spectral gap A1 > 0. Therefore, by Theorem 5.1.8, (5.1.1) holds for some C1 > 0 and C2 - 0 . (c) If (5.1.1) holds then by (5.4.4),
IIf211 ~ ~ C18(f, f) + C2 + c,
# ( f 2 ) _ 1.
(5.4.6)
f[E~ -
0, by Jensen's
Let Sn "= #(End). For any f with #(f2) = 1 and inequality on E~ we have
#(f2 log(1 + f2/A)) /> log(1 + 1/(Aen)).
234
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Then [[f2[[e ~> inf {A > 0" )~-1 log(1 + 1/(AEn)) < 1} -" An. Since an --* 0 we have An ~ oc as n ~ c~. Thus, there exists n ~> 1 such that An~2/> 6'2 + c and hence, (5.4.6) implies
[[f2[[ ~ ~ 2 C 1 8 ( f , f),
#(f2) = 1, f I E n
-- O.
This implies 5n < oc (and hence 50 < oc) according to Proposition 5.4.2, (5.4.3), (5.4.4). Finally, if (5.1.6) holds then (5.4.5) reduces to
Ilf211 ~ ~< r S ( f , f ) + 3(r) + c,
p(f2) _ 1, r > O.
(5.4.7)
For any r > 0, there exists n ~> 1 such that An >~ 2(/~(r)+ c). Then (5.4.3) and (5.4.7) implies
Ilf 2 II ~ < 2rE(f, f),
flE.
-- O,
#(f2) _ 1.
m
By Proposition 5.4.2 and (5.4.4) we obtain 5n ~ cBn <<. 2cr. Since r > 0 is arbitrary we conclude that 5n --+ 0 a~ n --, oc. K]
5.5
L o g a r i t h m i c S o b o l e v inequalities for oned i m e n s i o n a l diffusion p r o c e s s e s
d2 d Let E be either [0, c~) or ( - c o , oc), consider L "- a(x)-~x2 + b(x)-d-~ as in w
Recall that C ( x ) " -
let
x b(r) _ e C(x) a(r) dr and #(dx) 9 a(x-----~dx. For any n ~> 0
fo
(~+n " ~--- sup {#([x, oo))log #(Ix, o~)) -1 } fn x e-C(y)dy, ~[n,~) 5-n" =
sup
{ # ( ( - o c , x])log # ( ( - o c , X]) -1 }
fXmne-C(y)dy.
T h e o r e m 5.5.1 (1) Let E = [0, oc). (5.1.1) holds for some C1, C2 > 0 (hence for some C1 > 0 and C2 -- O) if and only if 5+0 < oe, while (5.1.6) holds for some/3 if and only if 5+n ~ 0 as n ~ oc. (2) Let E - I~. (5.1.1) holds for some C1, C2 > O(and hence for some C1 > 0 and C2 - O) if and only if 5-0 + 5+o < c~, while (5.1.6) holds for some /3 if and only if 5-0 + 5+o --* O, as n ~ oc.
5.5
Logarithmic Sobolev inequalities for one-dimensional diffusion processes
235
Proof. We only proof (1) since the proof of (2) is similar. To this end, let us apply the weighted Hardy inequality. Let ~ be a class of nonnegative measurable functions on E. For any n ~> 0 define Bn "- sup IIl[r,~)[l~ ~n r e -C(z)dx. r>n
Let An be the smallest positive constant such that []f2[[~ ~< An#(af,2),
f ~ C~([0, co)), f[[0,n] - 0.
It follows from Proposition 1.4.1 and the proof of Proposition 4.4.2 that
Bn <~.An <. 4Bn.
(5.5.1)
Then the proof of Theorem 4.4.3 implies the sufficiency, i.e. (5.1.1) (resp. (5.1.6)) implies 6+0 < oc, (resp. 6+, --~ 0 as n ~ oc). Conversely, by (4.5.1) and the proof of the Theorem 4.4.3, there exists a constant c > 0 such that
#(f2 log+ f2) <~ C6+np(af,2) + c,
f e C~[0, oc), #(f2) _ 1, f[[0,n] -- 0.
(5.5.2) Since the Neuman heat kernel of L on [0, n] is bounded, there holds a local super log-Sobolev inequality according to Theorem 5.1.7, i.e. there exists /~n: (0, (N3) ~ (0, (:X3) such that #(l[0,n+l]f 2 log f2) < r#(l[o,n+l]aff2) +/3n(r),
(5.5.3) f e C 1([0, n + 1]), #(f21[0,n+l]) - 1. Let h ( r ) " - ( r - n ) + A 1. For any f e C~[0, c~) with p(f2) _ 1, it follows from (5.5.2) and (5.5.3) that #(f2 log f2) _ #(1[0,n+1]f2 log f2) + #((hf)2 log+ (hf)2)
<~ 2C6np(af '2) + 2C6n#(af21[O,n+l]) + c + rp(l[o,n+l]af '2) + ~n(r) (2C6n + r)p(af 2) + c(n, r), r > O, n ~ O. where c(n, r) = c + flu(r) + 2C6n sup a. Therefore, if 60 < oc then (5.1.1)
[0,n+l]
holds for some C1, 6'2 > 0 and hence, for some C1 > 0 and 6'2 = 0 since 60 < oc also implies A1 > 0 according to Theorem 1.4.3. If 6+n ~ 0 as n ~ oc then (5.1.6) holds for f l ( r ) : = inf{c(n, s ) : 2C5n + s < r}. [:]
236
Chapter 5
5.6
Estimates
Log-Sobolev Inequalities and Semigroup Properties
of the log-Sobolev
constant
on
manifolds Throughout this section, we assume that M is a complete Riemannian manifold with boundary OM either empty or convex. Let L := A + VV for some V E C2(M) such that C(V) "- / e V(x) dx < c~, where dx is the . /
volume measure. Let # ( d x ) " - C(y)-leY(X)dx and 8 ( f , g ) " - #({Vf, Vg}) with ~ ( 8 ) : - H2,1(#). We first introduce some equivalent descriptions of the curvature condition (2.1.12), which in particular imply Bakry-Emery's criterion, then present explicit estimates of the log-Sobolev constant by means of this criterion, the dimension-free Harnack inequality (2.5.3) and the coupling method. 5.6.1
Equivalent s t a t e m e n t s for the curvature c o n d i t i o n
Let M be a complete connected Riemannian manifold with OM either convex or empty. Let L := A + Z for some Cl-vector field Z. Let Pt be the corresponding (reflecting if OM # 0) diffusion semigroup. Next, let p be the Riemannian distance on M. For any #, u E ~ ( M ) , the set of all probability measures on M, define the LP-Wasserstein distance WpP(#, v) "-
inf 71"(pP)l/p, ~e*(v,~)
p ~ 1,
where ~ ( # , v) is the set of all couplings of # and v.
Theorem 5.6.1
The following assertions are equivalent to each other:
(1) (2.1.12)holds. (2) [VPtfl <<.eKztptlv f[, (3) IVPtf[ 2 <<.e2KztPtlV fl2, (4) Ptf 2 - ( p , f ) 2 ~<
(5) p, f2 _ (p,f)2 ~>
t >>.O, f E C~(M). t >>.O, f E C~(M).
e2 K z t - 1 Kz Pt]V f]2' 1 - e -2KZt
Kz
t >t O, f E CI(M).
IvP, SI 2, t ~> 0,
Se
C~(M).
(6) Pt(f 2 log f 2 ) _ (ptf2) log(ptf2) <. 2( e2KZ* - 1) p,]v f[ 2
Kz
t~>0, f E
C~(M). (7) (Ptf)(Pt(f log f ) - ( P t f ) log(Ptf)} >i
1 - e -2Kzt 2gz IVPtfl2'
t>~0, f E
C~ (M), f>~0. (8) (2.5.3) holds for all a > 1, t >1 0 and f E Cb(M). Moreover, for each p E (1, 2) they are also equivalent to each of the follows:
t~
o
.~
o
~0 o
9
~0~
o~
o~
'~
-I
I
q
o~
I%.
~
A\
I>
.~
A\A\
~~
"~ ;~
"
v/~~ ~ "-~
~
~
~
~
;~ ~
~
''
9
n~ , ~ ~ ' ~
~"
..~~~
0 ;~ . =
~
~ t~~~~
~ ~
~~~o
~6
oo
~TK~
~
~
~-~
~
~
~
0 oo
~o
r
+
~
r~
r~
0
0
v
o
o
~0
~.~ o
,---,
I
I @
~
i
~
cC
i
+
Jl ~
~
~
~
tl
~ ~
~
V/
~
m~
v
'~'
--~
,
m
r
~
~,,.~
o -~
~ ~
~
%
V/
A
~ ~ ~.~
,
II ""
~
o ~
@
~
b.0 n::~
.=l
9 ,--~
e
~o
@
-t-
!
~I
~
~
0
O
I
!
I
O
~
r
'
"~:
'
~
o
I ~"~
o
~
~
I
~A -~
II
@
,-Q
A
O
,,,.,,
:~
UJ
~
o
~
O
0
o~
o
"
9~ ~
..
"~"
I
O
~
A,,
"'~
A~
A~
o~
~'~
V/
238
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
2p
Letting q - 3 p - 2' by (5.6.1) and Hhlder's inequality we obtain
d
]
--~sPs(Pt_sfP) 2/p ds
Ptf 2 - (PtfP) 2/p -
_ 2 ( 2 - p) foot
--
p2
2(2-p) p2
[VPt_sfP[ 2 Ps (Pt_sfP)2(p_l)/p ds
~oote2K(t_s) (Pt_~lVfPlq)2/q Ps (pt_sfp)2(p_i)/pdS
~< 2(2 - p) ~0 t e2g(t_s)PtlV fl2d s = (2 - P)(e2gtK -- 1) PtlV fl 2. Therefore (9) holds for all f e C~(M) by a standard approximation argument. On the other hand, if (9) holds then for any positive f E C ~ ( M ) we have
h(t) "=
K[Ptf 2 --(ptfP)2/P] ( 2 - p ) ( e 2Kt - 1) <<"PtlV f[2"
Obviously, h ( 0 ) : = lim h(t)= IV/I 2. Then t---~0+
LIVfl2 _ ( d
lim h(t) - h(O) -~PtlV fl2) lt=o) t-,o+ t
>I 2
K
lim d { p t f 2 - ( p t f p ) 2 / P }
-- p t-,O+
-~
e2Kt - -
-
1
K
lim (e2ggt( t ) ) 2 '
2 -- p t-~O+
-- 1
where, by Taylor's formula,
g(t) "-(e 2Kt- 1)[PtLf 2- 2 (pt fP) (2_p)/PPtL fP] _ 2Ke2Kt [pt f2 _ (pt fP)2/p] P = (2gt + 2K2t 2 + o(t))[Lf 2 -+-tL2f 2 - 2f2-pLfP 2tf2-pL2yp p p -- p)t f2( l--p) (Lfp)2 + o(t)] - 2 ( 2 p2 J
- 2K(1 + 2Kt + o(t))[tLy 2 - 2ty2-PLyP
P + 2t2L2f2" _ (2-p2P)t2f2(l_p)(Lfp)2 - t2f2-pL2fpp = t2K{ - 4(2 - p)KIVf[ 2 + L2f2
+~
2(2 - p) f2(1-p)(Lfp)2 p2
2 fz_pL2f p + SK f2_pLfp _ 4KLf2} + o(t2 ) P P = t2K{ - 4(2 - p)K[Vf[ 2 + 4(2 - p)(Vf, V L f ) + 2(2 - p)LlVf[ 2 -[- C1 ( p ) f - l l v f[2n f + c2(p)f-21V f[ 4 q- c3(p)f -1 (V f, VlVfl 2) } + o(t 2)
5.6
Estimates of the log-Sobolev constant on manifolds
239
for some Cl (p), c2(p), c3 (p) e R. Therefore, replacing f by f + n we obtain 1LIVf[2 L[V fl 2 = L I V ( f + n)l 2 > - K [ V fl 2 + {V f, V L f) + -~
c2(p)[V fl 4 + cl(p)lV f[2n f + c3(p)(V f , V[Vf[ 2) + 4(2 - p)(f + n) 4(2 - p)(f + n) 2" Letting n ~ oc we arrive at 1
LIV fl 2 ) - g ] v f[ 2 + (V f, V L f} + -~L[V fl 2, and hence (2.1.12) holds. (2.1.12) r (10)" Let t > 0,p e (1,2) and positive function f e C ~ ( M ) be 2p fixed. Let q - 3 p - 2 By H51der's inequality we have
(Ps[VPt_sflq) 2/q <. (Ptf)2(P-1)/PPs((Pt_sf) 2(1-p)/p VPt_sfl2). Since (2.1.12) implies (5.6.1), it follows that
/o
_
-
p) [tao 1),{(Pt-sf)2(1-P)/P[VPt-sfl2} p2
2(2-
ds
2(2 - p) ~ t (ps]VPt_sflq)2/qds ) p2(ptf)2(p-1)/p 2(2 - p) ft (2 - p)(1 - e-2Kt)[vPtfl2 e-2K~ [vPt f l2ds ) P2(Ptf)2(P-1)/P Jo p2K(Ptf)2(P-1)/P Hence (10) follows from (2.1.12). On the other hand, if (10) holds then for any positive f E C ~ ( M ) ,
IVPtf[ 2 <<.h(t) "- p2K[(Ptf)2(p-1)/pPtf2/P- (Ptf)2] (2 - p)(1
-
,
t > 0.
It is easy to check that lim h(t) - [ V f l 2, then t---~0+
2{VL f , V f) ~ lim h'(t) - Kp2 lim t--~0+
2 - p t~0+ (1
g(t)
- e-2Kt) 2'
where by Taylor's formula,
g(t) "= ( 1 - e -2Kt) [ 2 ( p - 1)(ptf)(p_2)/p(ptf2/p)PtLf+(ptf)2(p_l)/PPtLf2/p P
- 2(Ptf)PtLfl - 2Ke -2Kt [(ptf)2(p-il/pptf2/p _ (Ptf) 2]
240
Chapter 5
= t2K{
4(2 - ; ) K
p2
Log-Sobolev Inequalities and Semigroup Properties
IVfl 2 +
-}- Cl (p) [V f[2L f f
+ c2(p)
4(2 - p ) 2 ( 2
p2
(V f,
-p)
p2
V[V fl 2) + c3(p)]V~ ]4~ } f
__
LlVf
12
+ o(t 2)
for some cx(p), c2(p), c3(p) E R. Therefore, (2.1.12) holds as observed above. (1) =~ (11)" For any x, y E M, let I~t 'y be the distribution of (xt, yt), which is constructed as the coupling by parallel displacement of the (reflecting) Ldiffusion process with (x0, Y0) = (x, y), see Chapter 2 for details. By the proof of Proposition 2.5.1 we have p(xt, Yt) <<.eKtp(x,y) a.s. Next, for any zr E c~(#, v), we have
7rt
" - -
/M x M I?t'YTr(dx, dy) e ~(#Pt, ~Pt).
Hence,
WP(#Pt'IgPt)P ~ /MxM pPdTrt < ePKztTr(pP). This implies (11). (12) =~ (3)" For any t > 0, let 7rt 'y be the coupling of ~xPt and ~yPt such that 7rt,y(p2) < e2Ktp(x, y)2. (5.6.2) Then for any f e C~ (M),
]P, f (x) - P,f(y)l 2 ~< (f if(z1)
-
p(x,y)2
f(z2)]Trt'Y(dzl,dz2) 2
P(x'y)2
e2Ktf
(5.6.3)
If (Zl)p(zll--z2) 2f(z2)12zr~'y(dzl , dz2 ).
Since f E C~(M), for any compact set D C M and any r > 0, there exists > 0 such that for any zx E D and any z2 with p(ZX,z2) ~ (~, one has
[f (zl ) - f (z2)[ 2
p(Zl,Z2)2
~< [Vf(zl)[ 2 + r
(5.6.4)
It follows from (5.6.2), (5.6.3) and (5.6.4) that
IYtf(x) -
y~f(y)[2 <<.e2Kt{ptIV f(x)[2 +
p(x,y)2
q-[]Vf[[27rt'Y({(Zl,Z2)'Zl ~ D or p(zl,z2) > 5})} < e ~ * { P , IVf(~)l~ + + IlVfll ~P~ I D~(x) + p(x, y) 2e2Kt /~2 }. first letting y --+ x then ~ --+ 0 and D --+ M, we prove (3).
K]
5.6
Estimates of the log-Sobolev constant on manifolds
5.6.2
241
Estimates of a(V) using Bakry-Emery's criterion
Let M be a complete connected Riemannian manifold with OM either empty or convex. Consider L "- A + VV for some V E C2(M) with C(V) "=
Mevdx < oc. Let #(dx) := C(V)-leVdx. Let K(V,.) be a function on M such that Ric(X, X)
Hessv(X, X) ) - K ( V , x),
-
x E M , X E TxM, IX[ = 1.
We call
a(V) "- inf{2p([Vf[2) 9 #(f2 log f2) _ #(f2) log #(f2) _ 1,
f E C~(M)}
the log-Sobolev constant of L.
Theorem 5.6.2
(Bakry-Emery's Criterion)
a(V) >1 -Kv := infx{-K(V,
x)}. Proof.
By Theorem 5.6.1 we have
Pt(f 2 log f2) <~
2(e 2KVt- 1) PtlV fl 2 + (Ptf 2) log(Ptf2), Kv
t >O, f e C ~ ( M ) .
Taking integral for both sides with respect to p we arrive at 2(e2Kv t #(f2 log f2) <~
Kv
-1)
~([V fl2)+#((Ptf 2) log(Ptf2)),
t>O,f eC~(M).
Since Ptf 2 ---+#(f2) in L2(#) as t --+ cr (see e.g. w and w below), and noting that Ptf 2 is bounded by ]]f]]~, it follows that #((Ptf 2) log Ptf 2) ---+ #(f2) log #(f2) as t --- cr Thus, we finish the proof by letting t ~ cr in the above inequality. C] Obviously, Bakry-Emery's criterion makes sense only when Kv < 0. To apply this result to more general case, we make use of the following comparison between log-Sobolev constants for different potentials V and U:
a(V) ~ a(U) exp[-6(V - U)],
(5.6.5)
where 6(V) = sup V - inf V. To check (5.6.5), we shall use the identity
/ M f l O g p f f ) d # - inf/Mt>O { f log f - f log t - f + t } d# for all strictly positive and smooth f and hence, the integrand on the righthand side is non-negative for all t > 0.
242
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Let p be the distance function to a fixed point o E M, and let /3(r) "infp>~r{-K(V, x)}. Obviously, 13(r) is increasing in r and 13(0) - infr)0 ~(r) - K v . For fixed_ k ) 0, define f ( r ) : = r if k - 0 and f ( r ) ' - sin ( x / ~ r ) / v / k if k > 0. Set ~ ( r ) " inf ~(u)/f'(u). As usual, 1 / V ~ is understood
~:f (u)e[r,~/ (2v~ 1)
as c~ when k - 0. Note that 13(r) - / 3 ( r ) when k - 0 since/3 is an increasing function. Finally, for fixed a E [0, I t / ( 2 v ~ ) ) , define
1 fl(r) "y(r) := f ( r ) Jo
~
~.
#(u)du,
r < 2v/~ , (5.6.6)
Fo
.-
r~>0.
-
Jo
Jo
5.6.3 Assume that M ~ c u t ( o ) = 0 and the sectional curvature of M is bounded above by a constant k E I~. (1) Let k - O. If SUPra>0 Z(r) > 0 then
Theorem
[ /o
a(V) >~ a---~oexp 1 -
r~(r)dr] > O,
where ao > 0 is the unique solution to the equation
~Oa
(5.6.7)
~ ( r ) d r - 2/a.
(2) Let k > O. If cut(o) ~ S (o, 7r/(2v/k)) - O and ~/(a) > 0 for some a e (0, Tr/(2x/~)), then a(V) >1 f'(a)~/(a)exp[-Fa(a)] > O.
Proof.
By Proposition 2.1.5, when OM is convex we may regard M as a convex domain in a complete Riemannian manifold. By the Hessian comparison theorem we have (see w
(5.6.8)
Hessp(X,X) >~ ( f ' / f ) ( p ( x ) ) ( 1 - (Xp(x))2), where r f(r)'=
if k - 0,
sinh ( ~
r)/~
sin ( x / ~ r ) / v / k
if k < 0, if k E ( 0 , 7 r / ( 2 ~ ) ) .
The proof of part (1) consists of the following four steps. (a) Let supr~>0~(r ) > 0. Then, we have/3(0) > - c ~ .
f(r)-
r, it follows from (5.6.6) that ~ ( r ) -
and F~(r) =
/0 /0 ds
r1
/0
Since k = 0 and
fl(s)ds, r > 0, ~/(0)
fl(0)
Ca(u)du for C~(r) "- [~/(a)-~(r)]l[r<~a], a,r ) O. Note
5.6
Estimates of the log-Sobolev constant on manifolds
243
t h a t 13(r) is increasing in r and so is v(r). Next, let a(a) := 7 ( a ) e x p [ - F a ( a ) ] for simplicity. We will prove the following two assertions:
a(V) >~sup G(a)
and
a/>0
G(a)
sup
= G(a0) ,
where a0 > 0 is d e t e r m i n e d uniquely by the equation
a(V) >~ G(ao).
imply T h e o r e m 5.6.3 (1), i.e. assertion. Note t h a t
Fa(a) - j f o adr ~ o r Iv(a) a2
= -~-v(a)-
a
/0
(5.6.9)
a~0
/3(r)dr -
2/a,
which
Let us first prove the second
- / 3 ( s ) ] d s - -~-v(a) a2 1
7(r)d
j~oa
j f o ad~r o r fl(s)ds
r2,~ ~
- ~
(r)dr.
a
G'(a) = 7 ' ( a ) [ 1 - a2v(a)/2] exp[-Fa(a)]. Because a~v(ao), we have G' >~ 0 on [0, a0] and G' ~< 0 on.[a0, c~).
Hence
V' >~ 0 and 2 Thus, the global m a x i m u m of G is achieved at a0. This proves the second assertion in (5.6.9). Next, since a~v(a0) - 2, we have
a2
~176
r
(co - s)13(s)ds
=
Thus,
1 -
ag
G(ao) coincides
(ao) +
/o oo
-
-1
+
/ooor~(r)dr.
w i t h the lower b o u n d given in (5.6.7). Moreover,
G(ao) =
s u p G ( a ) >~ G(0) =/3(0).
(5.6.10)
a>~0
(b) We now begin to prove the first assertion in (5.6.9). Since a ( V ) ~> 0, we need only to prove f o r the case where sup~> 0 -y(a) > 0 (equivalently, supra> 0 ~ ( r ) > 0). Because
Eta (r) --
Ca(u)du - r
a
13(u)du - r
/3(u)du
,
r < a,
we see t h a t F~a(r) >~ 0 i f r < a and F~(r) = 0 i f r ~> a. Hence, 5(Fa) = sup Fa - inf Fa = sup F~ = F~ (a). Next, since Ca (a) = 7(a) - ~(a) ~< 0, Ca m a y not be continuous at a. For this, we need a modification of Ca. Let C (0, a) and define
CEa(r ) ' -
Ca (r) - Ca (a) r - r
if r E [0, ~],
Ca(a)(1
if r C [ a , a + r
C~(r)
r-a~r
otherwise.
244
Chapter 5
Let F~a(r) "-
Log-Sobolev Inequalities and Semigroup Properties
fo r ds f0 s C~(u)du.
Then Ca~ e C(R+) and Fa~ e C2(I~+).
Moreover, it is not difficult to check that (Fa~)' >~ 0, (F~a)'(r) - 0 for all r ~> a + e and C~(r) - -~ 1
j~0r C~(~)d~
~< o (Note that
jr0a C a ( ~ ) d ~ -
0). Hence
~(Fg) - sup Fg - F2(~ + ~) --, Fa(~) as ~ --, 0. (c) Take ~ ( x ) - F~(p(x)). Then 5 ( ~ ) = Fg(a + e). On the other hand, for x E M and X E T x M with I X [ - 1, by (5.6.8) we have
Hessy~ (X, X ) - (F~a) ' (p)Hessp(X, X ) + (F~a) '' (p)(Xp) 2
1/0
~> -p
C~ (u)du +
[
C~ (p) - P
]
Ca~(u)du (Xp) 2 >t C~ (p).
Here in the last step, we have used the facts that (Xp) 2 ~< ][X[[2 - 1 and C~ (p) - P
C~ (u)du<~ 0. Therefore, inf K ( V -
xE M
V~,x) >~ inf{Ca~(r) + ~(r)} >/~(a). r >~O
By Theorem 5.6.2 and (5.6.5) we obtain c~(Y) ~> ~(a)exp[-Fa~(a + e)]. Then the desired inequality follows by letting ~ --, 0 and hence we finish the prove of (1). (d) Finally, the proof of part (2) is similar. Recall that the functions ~ and F~ are given by (5.6.6) with f ( r ) " - sin (x/~r)/v/-k. By using the smoothing approximation as above, we may and will assume that F~ is a C2-function. Next, for p(x) < a, we have F~a~(p) = f'(p)[~(a) - ~ o/(p)] ~> f'(a)7(a) - ~(p). Thus, as we did above,
HessF~(p)(X,X) - F~a(p)Hessp(X,X) + F~(p)(Xp) 2 ~ f'(a)7(a) - ~(p). Therefore, K ( V - F ~ ( p ) , x ) >t f'(a)7(a) for p(x) < a. On the other hand, since Fa(p) - F~(~) fo~ all p > ~, we have K ( V - Fa(p),z) - K(V,x) >1 ~(a) >i f'(a)~ o f(a) >i f'(a)7(a) for p(x) >1 a. Thus, the desired conclusion follows from Theorem 5.6.2 and (5.6.5). F3
5.6.3
Estimates of a(V) using Harnack inequality
In this subsection we assume that M is compact with diameter D. We will estimate c~(V) in terms of the Harnack inequality of the so-cMled log-Sobolev function. A function f is called a log-Sobolev function (abbrev. LSF) if it achieves the log-Sobolev constant i.e. f is non-constant with #(f2) _ 1 such 2#(IV f[ 2) that c~(Y) = #(f2 log f2), where # is the probability measure on M parallel to eYdx.
5.6
Estimates of the log-Sobolev constant on manifolds
245
Following the arguments of [60], [163], a LSF is a solution to
Lf-
- - -a(V) ~-f
log
f2 .
(5.6.11)
Conversely, a nonconstant normalized solution to (5.6.11) is a LSF. Since any LSF does not have zero point, without loss of generality, we only consider positive LSF's. We assume that (2.1.12) holds for Z := VV, i.e. Ric-Hessv is bounded below by - K v . When V = 0 we simply write K and a instead of K0 and a(0) respectively.
Harnack inequalities for log-Sobolev functions Theorem
Suppose that V = 0 and OM = 0. If f > 0 solves (5.6.11)
5.6.4
then IV log
f[2 + (a + rg) log f ~< d(2 -16ar(1 r)2(a - +r)rK) 2 '
supf<~ inf exp[d(2-r)2(a+rK)]
[d
16ar(1 - r)
rE(0,1)
Proof.
Let r "- log f and r "- l y e ] 2 + determined. By (5.6.11) we have -
-IVr
-
e
r
(0, 1),
dK]
~< exp ~ + ~
.
(5.6.12)
(5.6.13)
(a + rK)r where r c (0, 1) is to be -
-r
+
Krr
Let x0 be the maximum point of r then at x0 we have 0 ~> A r - 2 ~ r i,j
2
/> ~ ( A r
2r
--d
+ 2~
2 - 2KIVr
-
Cj(Ar
+ 2Ric(Vr r e )
+ (a + r K ) A r
j 2 + 2KrlVr
{ 4Kr d r
2 - (a +
}
Kr)r + Kr(a + Kr)~2
2K2r2r + K(a + Kr)(2_ r)~2,
r
where for a smooth function F, Fi and Fij a r e the first order and the second order partial derivatives under a normal coordinate frame at x0. Let s :--
Kda(r - 1)r + T~[K(2 - r) + a] 2, we obtain at xo that d [K(2 -
r _
+
+
rd
d
-- - (1 -
r)da
16(1
r(K(2-r)+a)] 2
~< d(l - r) [
4ra
4
a+
2(1 - r)
"
-
r)a
Chapter 5
246
Log-Sobolev Inequalities and Semigroup Properties
This proves (5.6.12). Finally, let y0 be the maximum point of f, by (5.6.12) we have (a + rg)log f(Yo) = r
< r
~< d(2 - r)2(a + r g ) 2 16ar(1 - r) " This implies the first inequality in (5.6.13) since a + r K > 0 for r E [0, 1] according to the known estimate (see [73]) {A1 3A1- d K } a/>max ~-K, d+2 >-K, where A1 > 0 is the first eigenvalue of the Laplace operator. Then the second inequality of (5.6.13) follows by taking r = 2/3. [2 Under the assumption of Theorem 5.6.4. If f > 0 solves
Corollary 5.6.5 (5.6.11) then
f (X) < f (y)1-e exp
a + --
P
+ ~aa
'
x, y e M, e e (0, 1). (5.6.14)
Proof.
By (5,6,12) with r = 2/3 we have d (a § 2K)2 IVlogfl ~
( a + 2 K ) log f. 5
(5.6.15)
For any x, y E M with p(x, y) > 0, let -y" [0, p(x, y)] -~ M be any minimal geodesic from x to y. Choose sl and s2 such that
f(')'Sl)-- max f('Ts), [0,p(~,u)]
f('Ts2)--
min f(%). [0,p(~,y)]
(5.6.16)
2 Noting that a + ~ K > 0 as observed above, by (5.6.15) we obtain
~(~~ log-
~x d
(a + ~K) log f (Ts2).
d (c~+ Therefore,
f (~) ~ f (7~1) ~ f (7~) exp p(x, v)
~
~< fl-s(%~)exp [e log f(%2) I-
d (c~+
+ P(~' Y) ~
-5
-
2 )2 (,~+g2 K ) log f(')'s2)]
~ +-~
(o +
5.6
247
Estimates of the log-Sobolev constant on manifolds
Let s "-
~/ ~
2
g
-(~+5
2 K) log f (%2), we obtain C8 2
f ( x ) <<.f l - e ( y ) exp [ -
2
+ p(x, y)~ + ~
g
a + -~K < fl-e(y)exp
[
(a + 5
D
4e
T h e o r e m 5.6.6 Suppose that V E C2(M) and OM is either convex or empty. If f > 0 solves (5.6.11) then (sup
f)l-2a(V)t ~
(inf f)l-a(V)t exp
KvD2 ] 1 - exp(-2Kvt) '
t e (0, ( 2 ~ ( v ) ) - l ) .
(5.6.17) Proof. Let x0 and Y0 be respectively the maximum point and the minimum point of f. By (5.6.11) we have Ptf(xo) - P~f(xo) -
EZ~ f(x~)du - - a ( V )
>f -~(v)log f(xo)
E~~ f(xu) log f(xu)du
P~f(xo)d~,
t >/~ >/O.
This implies
(5.6.18)
Ptf(xo) >~ f ( x o ) e x p [ - a ( V ) t log f(x0)] - fl-a(Y)t(xo). Similarly, we have
Ptf(Yo) <~ fl-a(v)t(yo).
(5.6.19)
On the other hand, by (2.5.3)
(Ptf(xo)) 2 <~ ptf2(yo)exp
[ vo2 ] [ vo2 ] e -2Kvt <<"f(xo)Ptf(Yo)exp e -2gvt " 1-
1-
(5.6.20) The proof is now completed by combining this with (5.6.18) and (5.6.19). 73
Corollary 5.6.7
Under the assumption of Theorem 5.6.6. If f > 0 is the
LSF, then sup log f <~D2a(V) + D2K +, inf log f ~> -
27 D2 a(V) - 9 D2K+ I--6 8 v"
(5.6.21)
(5.6.22)
248
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Proof. Let x0 and Y0 be, respectively, the maximum and minimum points of f. By (2.5.3) and (5.6.18), (5.6.20) holds for any y in place of y0. Since # ( p t f 2 ) _ #(f2) _ 1, we obtain KvD 2 log f (xo) <<. 2(1 - a(V)t)(1 - e-2Kvt) '
a ( V ) t e (0, 1).
Next, it is easy to check that Kv 1 - 2 g v t <<"~
1 -e
t>o.
+ K+'
(5.6.23)
Then by taking we obtain 1 D 2 ( ~ + K +) log f(xo) <<. 2(1 - a'--'~'~v)~) = D2a + D 2 K +,
t -
2.(v)"
Next, By (5.6.18), (5.6.19) and (2.5.3) we obtain 1 <~ f l - S a ( V ) t ( x o )
By taking 5 =
5KvD 2
4(5- 1)(1- e -2Kvt)
<~ f l - a ( v ) t ( y o ) e x p
(v)t
]
5a(V)t ~< 1.
we arrive at
-KvD 2 log f (Yo) >~ 2(1 - a(V)t)2(1 - e - 2 K y t ) '
. ( v ) t < i.
1 Combining this with (5.6.23) for t - 3a(V), we obtain (5.6.22). Estimates
K]
of a(V)
In case that the LSF may not exist, we turn to consider the defective logSobolev constant a~ for small ~ > 0, which is the largest possible constant such that #(f2 log f2) ~
2
+
f e CI(M), # ( f 2 ) _ 1.
Then a~ ~ a ( V ) as ~ ~ O. By [163], the constant a~ can be achieved and the corresponding LSF could not be constant since ~ ~ 0. Let f~ be such an LSF, then ([163]) Lf~ -
a~
1
5.6
249
Estimates of the log-Sobolev constant on manifolds
As mentioned above, we may assume that fe > 0 (cf. [60]). Therefore, by the proofs of Theorem 5.6.4 and Corollary 5.6.7, we have (5.6.25)
sup log fe <~ D2ae + D 2 K + + he, d dK sup log fe ~< ~ + ~ + h~
if Y - O, OM - 0;
(5.6.26)
where he --* 0 as ~ --, 0. T h e o r e m 5.6.8 Under the assumption of Theorem 5.6.6. Let A1 be the first eigenvalue (or the spectral gap) of L. We have V/(1 + D2K+) 2 + 4 D 2 A 1 - 1 - D 2 K + a(V) ) 20 2 .
(5.6.27)
Especially, if K v <~ 0 then a ( V ) >1
2.68 v/1 + 4D2A1 - 1 v/1 + 47r2 - 1 2D 2 >1 2D 2 > D2 .
(5.6.28)
Finally, for the case that V = 0 and OM = 0, we have
6A1 - 2dK a ( 0 ) ( = a ) ~> 3 ( d + 2 ) "
(5.6.29)
By the spectral representation we have
Proof.
(5.6.30)
# ( ( L f~) 2) >1 AI#([Vf~I2).
Next, by (5.6.24), 2
.((Lf~) ~) - ~.{4 = ~{4
2
)2
s
2
(log f~ } + -~ ae - r
ff log re}
~ (log f~ )~ } - ~.(IVf~
I~ )
(5.6.31)
~~4 "
Noting that ~oLe f2 feL(fc log fe) - - c~efs2 (logfe)2 + -~~ log f~ + f~Lfs + IVf~l 2
we obtain s
s
aep{ff(1ogfe) 2 } -- T~{f~
log f~} + ~{(1 + log f~)[vf~l2}
s
E
= ~ ~(Ivf~12) + E
4 + ~{(1 + log f~)Ivf~1~} \
~
\
/~[1+ ~ + sup log fe}#{]Vf~[ 2) + / .- .
E 20Ls
--T
250
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
By combining this with (5.6.30) and (5.6.31), we arrive at ~l#(Ivf~l 2) ~ c~(1 + sup log L) (IVLI2). This implies )~1 ac ~> 1 + sup log re"
(5.6.32)
By (5.6.25) and letting e --, 0, we obtain i>
)~1 1 + D2c~(V)+ D 2 K +"
7r2 D2 for K v <<. 0, see Corollary 2.2.2. Finally, (5.6.29) follows from (5.6.26) and (5.6.32). I-I
This proves (5.6.27), and (5.6.28) then follows from the fact that /~ t>
5.6.4
Estimates of a(V) using coupling
The Harnack inequalities proved above enable us to estimate c~(V) by using the coupling method. By the approximation procedure as in the last subsection, we may assume that the LSF exists. The coupling method then works as follows. T h e o r e m 5.6.9 Let f > 0 be the LSF. Define /~1 = sup f and /32 = sup I1 + log fl. Next, let (xt, Yt) be a coupling for the L-diffusion process with coupling time T = inf{t ~> 0: xt = yt}. We have E~,UT} -1 ~1 - 1 (1) oL(Y) ~> { SUPx'y~M ) fll log fll" (2) /f there exists ~ E C ( M • M) with ~ >~ cp for some c > 0 such that EZ'Yp(xt, yt) <~ f(x, y)e -~t
(5.6.33)
for some 5 > 0 and all t ~ O, Then c~(V) ~ 5/~2. Proof. (a) Let x0 and y0 be respectively the maximum point and the minimum point of f. We have 1 - f(xo)-
1
f(yo)
{
E~~176
f(yt)]-
/o E ~~176
[ L f ( x ~ ) - Lf(y~)]ds
~ ( Y ) 5 ( f log f) fo t <<.Px~176 > t) + f(xo) - f(Yo) " Px~176 > s)ds,
}
5.6
Estimates of the log-Sobolev constant on manifolds
251
where
5(f log f) := sup f log f - inf f log f = f(xo)log f(xo) -[f(Y0) V e -I] log[f(y0) V e -1] f(xo)
(1 + log s)ds f Jf(y0)ve-~
-
~ ( f ( X o ) - f(Yo) V e-l)/~1------~_1
(1 -Jr-10g 8)d8
/31 log/31
<. (f(xo)-
f(yo)) 5i - 1 "
1
Here, we have used the facts that f(Yo) ~< 1 and
/~1 -- r
increasing in r. Therefore, ~(v)
f/~1 (1 + log s)ds is Jr
(ill- 1)px~176 <~t)
t>
/~1 log/31Ex~176
This proves (1) by letting t ~ c~. (b) For any s > 0, choose x~ r y~ such that
f (xc) - f (y~) >~ sup f (x) - f (y) --E :-- C - - E . Noting that If(x) log f(x) - f(y)log f(y)[ ~< /32If(x) - f(y)[ ~ C~2,
~(x,y)
~(x,y)
by (5.6.33) we obtain ( C - ~)fi(x~, y~) < f(x~) - f(y~)
EX~'Y~lf(xt) - f(Yt)l + a
Ex~'Y~l(f log f)(x~ )
- (f log f)(ys)lds
CEX~'Y~fi(xt, yt) + aC~2
/o
Ex~'Y~fi(xs, y~)ds
~ Cfi(xe ye) [e-St + a/32(1- e-St)] '
(~
"
Then the proof of (2) is completed by letting t ~ c~ and s ~ 0.
[::]
252
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
By Theorem 5.6.9 with Kendall-Cranston's coupling, we obtain the following result. C o r o l l a r y 5.6.10
(1) If K v <<.0 then co 2.73 a ( V ) >i ~ > 0 2 ,
where co > 0 solves c 2 - 8 ( 1 - e-~). (2) Suppose that V = 0 and OM = 0 . If K <~ 0 then
16(1 dD 2 Proof.
Let F(r) "=
exp
- - ~ K v s 2 ds
exp -~Kvu 2 du, r >>.O. For
Kendall-Cranston's coupling we have (see Theorems 2.1.1 and 2.1.4) d F o p(xt, Yt) ~ dMt - 4dt 1 1 for a martingale Mr. This implies that E x'yT <<. --4F(D). Then E~,y T >~ 8 if g v <<. O. Thus, (2) follows from (5.6.13) and Theorem 5.6.9(1). Next, D2 by Corollary 5.6.7, ~1 ~ e a(V)D2. By Theorem 5.6.9 (1) we obtain a ( V ) >~ 8(1 - e -av2) 9 Letting c - D 2 a ( V ) we have c > 0 and c2 ~> 8 ( 1 - e - ~ ) . D4a(V) Therefore, c ~> co if co > 0 satisfies c~ - 8(1 - e -~~ It is easy to check that co > 2.73. Vl
5.7
Criteria
of hypercontractivity,
superbounded-
ness and ultraboundedness Let M be a d-dimensional, connected, noncompact, complete Riemannian manifold with boundary OM either convex or empty. Let Pt be the semigroup of the (reflecting) diffusion process generated by L - A + Z for some Cl-vector field Z satisfying (2.1.12). Assume that Pt has an invariant measure # which is positive and Radon. In particular, this is the case when Z = VV. 5.7.1
Some criteria
We first extend Theorem 5.1.4 to the present nonsymmetric case.
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
253
T h e o r e m 5.7.1 (1) Assume (2.1.12). /f there exist C, t > 0 and q > p > 1 such that [[Pt[[p~q <~ C, then #(f2 log f2) ~ 2(exp[2Kzt] - 1)p(q - 1) #(IVf[2) + pq log C Kz(q-p) q-p
(5.7.1)
holds for any f e C ~ ( M ) with #(f2) _ 1. (2) If there exist C1, C2 > 0 such that
p(f2 log f2) ~< Clp(lVfla) + Ca, then
f e C ~ ( M ) , # ( f 2 ) _ 1,
(5.7.2)
1 1 ,[Pt,[p__~q<~exp [ 4 C 2 ( p - q)]
for t > 0 and q > p > 1 satisfying exp[4t/C1] ~ (q - 1)/(p - 1). (3) Assume (2.1.12). Then Pt is superbounded if and only if (5.1.6) holds for some/3. Proof. It suffices to prove (1) since (2) follows immediately from Theorem 5.1.4 and Remark 5.1.1, and (3) is a consequence of (1) and (2). For the symmetric case, the key point of the proof (i.e. the proof of Theorem 5.1.4) is to use Stein's interpolation theorem and then apply Gross' theorem. This interpolation theorem, however, does not apply to our present case. Our proof is based on the following semigroup log-Sobolev inequality implied by Theorem 5.6.1: P t ( f 2 log f2) ~< 2(exp[2Kzt] - 1) p, lv fl 2 + (ptf2) log(ptf2) ' Kz
f C
(5.7.3)
which holds for all t > 0 under (2.1.12). Since (Ptf 2) log+(Ptf 2) ~< (Ptf2) 2 <~ ]]fll~(Ptf
2)
and f has compact support, it follows that (Ptf 2) log+(Ptf 2) e LI(#). Moreover, (5.7.3) also implies (Ptf 2) log-(Ptf 2) e L 1(#). So, since # is an invariant measure, (5.7.3) yields
#(f2 log f2) < 2(exp[2Kzt] - 1)
+ #((ptf2) log(ptf2)).
Kz
For any s e (0, ( p - 1)/p), let r -
ps p - 1'
1 Ps - ~ , 1- s
qs -
1 1 -5s'
5-
p ( q - 1) q ( p - 1)"
(5.7.4)
Chapter 5
254
Log-Sobolev Inequalities and Semigroup Properties
Then r E (0, 1) and 1 --
r --
Ps
- +
1 l - r ,
P
r
--
--
qs
- +
1-r.
q
By Riesz-Thorin's interpolation theorem, we have
IlPtllp~q~ ~ c ~ -
Cps/(p-1).
Therefore, for any f e C ~ ( M ) with # ( f 2 ) = 1,
f (Ptlfl2(1-~))q~d# ~ IlPtll ps---+qs q~ ~
C ps/(p-1)(1-Ss)
(5.7.5)
with equalities holding for s - 0. By the mean value theorem, for any s E (0, 1/45), there exists s ~ E [0, s] such that
[(ptlfl2(1-~))q~ - ( y t f 2 ) l 8
=[q's,(Ptlf[2(1-s')) q~' log Ptlf[ 2(1-~') - qs,[Ptlfl2(x-s')]q~'-lpt(lfl 2(1-s') log f2)[ <~[q's,(1
-
st)llf[[2[21-s')qJ-1] I(Ptf 2) log Ptf~l
nt- 2qs, llf[12(X-s')(qs'-x)+(X-2s')lPt(lf[ log Ill)l] <<.c([[fl[c~)[l(Ptf 2) log ptf2[ + [Pt([f[ log [f[)[] for some c(llf[[~ ) > O. Since (Ptf 2) log Ptf 2 and Pt(lfl log Ifl) are integrable, by the dominated convergence theorem we obtain from (5.7.5) that
d # ((pt[f[2(l_~) )q~) [s=O < P 5#(Ptf 2 log Ptf 2) _ # ( f 2 log f2) _ ~s p-1 By combining this with (5.7.4) we prove (5.7.1).
log C. V1
Theorem 5.7.1 implies the following interpolation theorem. C o r o l l a r y 5.7.2 q>p>l, then
Assume (2.1.12). If I[Pt[[p__,q <~ C for some t, C > 0 and [[Psl[p~-~q~ <~ C4Pq(q~-P~)/P~q~(q-P)
for any s > 0 and qs > ps > 1 satisfying s~
(exp[2Kzt]- 1 ) p ( q - 1)log q~ - 1 2Kz(q - p) p~ - 1
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
255
T h e o r e m 5.7.3 Assume (2.1.12) and let # be a probability measure. Let p be the Riemannian distance function from a fixed point o E M. We have (1) Pt is ultrabounded if and only if [[Pt exp[AP2]I[~ < c~ for any A,t > O. (2) Pt is superbounded if and only if #(exp[Ap2]) < c~ for any A > O. (3) If there exists A > g z / 2 such that #(exp[Ap2]) < c~, then [IPt[[2+4 < 1 for some t > O.
Proof.
By Theorem 5.6.1 and (2.1.12),
Kzp(x,y) 2 [P~f(x)]2 ~ P~f2(y)exp [ 1 - - e - ~ p i - 2 K s ] ] '
(5.7.6)
where p(x, y) is the Riemannian distance between x and y (recall that p ( x ) " p(x, o)). This implies that
1
i s(x)l
exp [
>1 IP~f(x)12#(S(o, 1))
1
Kzp(x,y)2 [ [-
Kz(p(x) + 1) 2 ] 1 - exp[-2g ]J'
where Bo(r) is the geodesic ball with center o and radius r. Then there exist Cl, ca > 0 such that
[p,f]
exp [(Cl -+-c2p2)18],
s e (0, 1].
(5.7.7)
Now we prove (1). By (2.7) we have IIP~II~-~ < IlPt/2 exp[2(Cl + c2p2)/t]ll~ < c~,
t e (0,1]
(5.7.8)
provided [[Pt exp[Ap2][l~ < c~ for any t,A > 0. On the other hand, if Pt is ultrabounded, then it is superbounded and thus (5.1.6) holds by Theorem 5.7.1(3). Hence exp[Ap 2] E L2(#) for any A > 0 according to Corollary 5.3.2. Therefore, [[Pt exp[AP2][[~ ~< [[Pt[[2~[[ exp[AP2][[2 < cxD for any t,A > 0. The proof of (1) is completed. Next, (2) follows from (5.7.7) and Corollary 5.3.2 below immediately since by Theorem 5.7.1(3) the superboundedness of Pt is equivalent to (5.1.6). Finally, if there exists A > K z / 2 such that p(exp[Ap2]) < c~, then there exists t > 0 and q > p ~> 2 such that [[Pt]]p+q < (:~. This can be proved by the argument leading to (5.7.7), just consider f with [If lip = 1 instead of ]]f]12 = 1, and apply the dimension-free Harnack inequality (2.5.2) with power p, see [200] for details. Then, by Riesz-Thorin's interpolation theorem, we have [IPt][2+q < c~ for some t > 0 and some q > 2. Therefore, (5.7.2) holds for
256
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
some C1, C2 > 0. Moreover, since # has a strictly positive continuous density with respect to the volume measure (see e.g. [26, Theorem 1.1(ii)]), according to Theorem 4.3.1 there holds a weak Poinca% inequality. Combining this with (5.7.2), Theorem 3.3.1 and Proposition 4.1.2, we conclude that the Poinca% inequality holds. Therefore, (5.7.2) holds indeed for some C1 > 0 and C2 = 0. Hence, the proof is completed by Theorem 5.7.1. [3
C o r o l l a r y 5.7.4 In the situation of Theorem 5.7.3. Let L be symmetric. Then [IPt[ll__.~ < oo for any t > 0 if and only if []Pt exp[AP2][[~ < oo for any A, t > 0. Furthermore, [[Yt[[1--+~ < exp[)~l/t][[Yt/4 exp[A2p2/t][[~ for
some
)~1, )~2 > O.
Proof. Since when Pt is symmetric one has I[PtIIl__.~ proof is completed by Theorem 5.7.3(1).
[IPt/21[2~, the [3
Theorem 5.7.3 enables us to present the following result which is still valid even when the Ricci curvature is not bounded below (see Example 5.7.1 below). C o r o l l a r y 5.7.5
Assume that (2.1.12) holds and there exists a strictly pos-
itive, increasing ~/ e C(O, oo) such that lim ~/(r) = oo, g~ (r) "- r~,(A log r) is r----~oo
r
convex on [1, c~) for any A > O, and Lp2(x) <~ c - ~/(p2(x)) holds for some c > 0 and all x ~ cut(o). probability measure #. If /2 ~ rT(Adrlog r) < oo,
(5.7.9)
Then Pt has a unique invariant
A > 0,
(5.7.10)
then Pt is ultrabounded. If in particular (5.7.9) holds for ~/(r) - ar ~, where a > O, 5 > 1 are fixed, then
IlPtll2-
exp [ct -5/(5-1)]
(5.7.11)
for some c > 0 and all t E (0, 1]. Proof. The existence and uniqueness of # follows from the proof of [26, Corollary 3.6]. Let xt be the L-diffusion process (with reflecting boundary if OM ~ 0). We have (by [117] and the assumption on OM)
d xp[ p
xp[ p 2(x )]db + l{xt~tcut(o)}Lexp[Ap2](xt)dt - dLt,
(5.7.12)
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
257
where bt is a one-dimensional Brownian motion and Lt is an increasing process with support contained in {t ~> 0 : xt C c u t ( o ) U O M } . Here and in what follows, the function L exp[Ap 2] is defined outside of cut(o). On the other hand, by (5.7.9) we have L exp[Ap 2] - A exp[Ap2]Lp 2 + 4A2p 2 exp[Ap 2] ~< A exp[Ap2] (c - 7(p 2) + 4Ap 2) < A exp[Ap 2] (el -- ,.y(p2)/2), wherecl-0Vsup
{ c - ~ 71( p
} 2)+4Ap 2 <
(5.7.ia) let h ( s ) ' -
oc. For f i x e d x E M ,
E z exp[Ap2(Xs)], s E [0, oc). By (5.7.13), Lexp[Ap 2] is bounded from above by some C(A) > 0 for each A > 0. By (5.7.12) and (5.7.13), E x exp[)~p2(Xs/~rn)] <<.exp[Ap2(x)] + C(A)EX[s A Tn], where ~'n = inf{t ~> 0 : p(xt) >~ n}. This implies that ~-~ ~ cc a.s. as n ~ oc, and hence h(s) <<.exp[Ap2(x)] + C(A)s for all A > 0 and s >~ 0. In particular, e ~p2(x~) is uniformly integrable so that h is continuous. Furthermore, Mt "-
fo
t2,kv~p(xs)exp[Ap2(x~)]db~
is & square integrable martingale for each A.
Moreover, we note that the one-dimensional Lebesgue measure of {t >~ 0 : xt e cut(o)} is a.s. zero since L is strictly elliptic and cut(o) has zero volume. By (5.7.12), (5.7.13) and Fubini's theorem, for any s ) 0 and s > 0 we have
h(s + s) - h(s)
e A cl
fs+
h(t)dt - -~cos
E ~ exp[Ap2(xt)]7(p2(xt))
h(t)dt -
h(t)3,(A -1 log h(t))dt
dt
s Js
since g~ is convex on [1, oc) for any A > 0. This implies A
h'(8) ~ ~clh(8) - ~ h ( 8 ) ~ ( ~ -1 log h ( 8 ) ) ,
s~>0.
(5.7.14)
Taking c2 ~> 1 such t h a t 7()~ -1 log c2) >~ 4Cl, we obtain A h(s)~/(A_ 1 log h(s)) < Acl h(s) - -~
-~ A h(s).y(A_ 1 log h(s))
if h(s) ) c2.
(5.7.15) By (5.7.14) we see that if h(so) <~ c2 for some so ~> 0, then h(s) <~ c2 for all s > so. Indeed, let te = inf{t ~> so : h(t) >>. c 2 + s } , it suffices to show that tc = cc for any s > 0. If there exists s > 0 such that t~ < oc, let t~ - sup{t c [s0,t~]" h(t) <. c2}, we have h'(s) ~< -ClA < 0 for s E [t~,t~],
258
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
hence by continuity h(te) < h(t~) <~c2 which is a contradiction. Now, let t > 0 be fixed. Therefore, if h(t) > c2 then h(s) > c2 for all s e [0, t]. By (5.7.14) and (5.7.15), we obtain A h(s).y(A_ 1 log h(s)), h'(s) <~ --~ By (5.7.10) ' we may define a(r) =
s e [0, t] if h(t) > c2 .
(5.7.16)
dr'log r ~) which is finite for r > 1. r~,(A -1
It then follows from (5.7.16) that, for the case h(t) > c2,
G(h(t))
L cr dr fh(O) dr A r')/()k -1 logr) ~> (t) Jh(t) r-~(A -1 log r) >~ ~t.
=
Therefore,
h(t) <<.G-I(At/4) V c2
(5.7.17)
for all t ~> 0, since G is strictly decreasing, where G -1 (r) "= 1 if r ~> sup(1,oo ) G. The first assertion then follows from Theorem 5.7.3 (1). If (5.7.9) and (5.7.10) hold for -y(r) - ar~(a > 0,6 > 1), then G(r) a(6-
I)
(log r) 1-~ and Cl
for some c3, C4
>
-0.
sup{c - 7(p2)/2 + 4Ap 2} = c3 + c4)~5/(5-1) Then, we may take
c2 -- exp [(4Cl/a)l/hA] < exp [c5 +
C6)~5/(5-1)]
for some c5, c6 > 0. By (5.7.17),
115/2exp[Ap 2] II~ < G-1 (At/S) V c2 ~< exp [ ( a ( 6 -
1)t/(8)~6-1))l/(1-6)V
This implies (5.7.11) by taking A = 2c2/t in (5.7.8).
(C5-I-C6)~6/(6-1))1. [-q
We may also obtain a necessary condition corresponding to (5.7.9). A function f is said to be compact if {f ~< r} is relatively compact for any rER. C o r o l l a r y 5.7.6 In the situation of Theorem 5. 7.3. If there exists a compact function ~ E C2(M), with (N, V/5} >~ 0 whenever OM r 0, where N is the inward normal vector field of OM, such that 15 > 1, IVv/~l ~< 1, and Lt5 ~> -c/5 log t5 for
c > O, th n
IIPtll2-
-
for
small t > O.
(5.7.18)
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
Proof.
259
For any n >~ 2, take hn C C~[0, c~) such that 0 <~ hn <<.1, hn(r) = 1
for r E [0, n], hn (r) - 0 for r >~ 2n, --2 <~ h~ <. O. Let f~(r) n
/0
hn(s)ds and
Yn - l+A(~xp[p]). We h~v~ IV~I 2 - 4 ~ L V ~ I 2 < 4~, ~ ~og ~ < 4 A ( ~ ) l o g A(~)
for r E [1, 2n] and rh~(r)/> - 4 . It then follows from (5.7.18) that
LFn - hn (exp[fi]) exp[fi]Lfi +
(hn
(exp[fi]) + h~n(exp[fi]) exp[fi]) exp[fi] IVfil 2
--l{exp[fi]<2n } (C log(2n) + 16) exp[fi]fi ~> -a(n)Fn log Fn,
(5.7.19) where a(n) = 4 ( c l o g ( 2 n ) + 16). By (5.7.3),
Pt(Fn log Fn) <<.(PtFn) log(PtFn) + 2 ( e x p [ 2 K t ] - l ) p t I V 'v~-In2, 1 K (PtFn) log(PtFn) + c2tPt(Fn log Fn) for some c2 > 0 and all t e [0, 1]. Let t e [0, 1/(2c2)], then
Pt(F~ log Fn) <~2(PtFn) log(PtF.). By combining this with (5.7.19), under the condition (N, Vfi) ~> 0 whenever OM ~= O, we obtain d d--~PtFn >~ -2a(n)(PtF~) log(PtFn),
t e [0, 1/(2c2)].
This implies PtFn >~exp {(log Fn)exp[-2a(n)t]} for t < 1/(2c2). Therefore, 1 + IIP~exp[p]ll~/> IIPtFnll~ ~ exp { (log n ) e x p [ - 8 t ( c l o g log(2n) + 16)]} = exp { exp[log log n - 8tcloglog(2n) - 128t]},
t ~< 1/(2c2).
By letting n --+ oc we obtain IlPt exp[filll~ = oc for small t > 0. The proof is now completed by Theorem 5.7.3(2) since v/-fi ~< V/fi(o)+ p. D Correspondingly to Corollary 5.7.6, we have the following result for superboundedness and hypercontractivity.
Assume (2.1.12). Let the limits below be taken outside of
C o r o l l a r y 5.7.7 cut(o). (1) If
lim p ( x ) ---+oc
Lp2 --~-(x) - -c~, then Pt has a unique invariant probability w
measure # and is superbounded. lim ~ -----~-(x) LP2 (2) If p(x)__+ < - 2 K z , then Pt has a unique invariant probability
measure # and is hypercontractive.
260
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Proof. The existence and uniqueness of # follows from the proof of [26, Corollary 3.6]. For A > 0, we have L exp[Ap 2] - A exp[Ap2](Lp2+ 4Ap 2) outside of cut(o). If plim - - * ~Lp2 - 7 < -4A, then there exist Cl, c2 > 0 such that L exp[Ap 2] ~< C l - C2 exp[Ap 2] outside of cut(o). Corollary 5.7.5, we obtain
By this and the proof of
Pt exp[AP 2] ~< Cl/C2 - } - e x p [ - c 2 t + Ap2].
(5.7.2o)
This implies that #(exp[Ap2]) <~ Cl/C2. Actually, let fin -- t9 A n for n ~> 1. It follows from (5.7.20) that #(exp[AP2n]) -- #(Pt exp[AP2n]) < #(exp[An 2] APt exp[AP2]) ~< #(exp[An 2] A (Cl/C2 + exp[-c2t + Ap2])). The desired assertion follows immediately by first letting t ~ oc then n ~ oc. The proof is now completed by Theorem 5.7.3(2) and (3). [2] R e m a r k 5.7.1 (i) From the proofs we see that Corollaries 5.7.5 and 5.7.7 remain true with p replaced by any smooth function ~ ~> Ap for some A > 0 such that (Vth, N)[OM <~ 0 whenever OM ~ O, where N denotes the inward normal vector field of OM. Corollary 5.7.7 (1) is sharp in the sense that Pt is not superbounded if OM - 0, o is a pole and plim - - . ~Lp2 - 7 > -c~. Actually, in this case #(exp[Ap2]) = c~ for big enough A (see the proof of [194, Corollary
1.4]). (ii) By Theorem 5.7.1, ][Pt]12---,4 < (:x:) f o r some t > 0 implies the defective log-Sobolev inequality for # which implies that there exist r, c(r) > 0 such that
#(f2) <~c(r)#(iV fl2),
f e C ~ ( M ) , f]Bo(~) - O.
Then the Poincar~ inequality holds (see [194]), and hence the log-Sobolev inequality holds (see also [2]) provided # is a probability measure. Furthermore, in this case the essential spectrum of L ~ := A + V log r is empty since L ~ is essentially self-adjoint on L2(#) (cf. [23]), where r
d# dx" Since Z is locally
for any p > 0. In particular, r is bounded, we have (by [23]) r c l/l/p'l(dx) ,, loc Hhlder continuous. Moreover, we see that L*r = 0 in the distribution sense. By [179, Corollaries 5.3 and 5.5], r is positive everywhere on M \ OM. (iii) By Theorem 5.7.3 (2) and (3), it is easy to construct examples to show that superboundedness is strictly stronger than hypercontractivity. Moreover, from Theorem 5.7.3 (2) and Corollary 5.7.6 we see that ultraboundedness is
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
261
strictly stronger than superboundedness (see Example 5.7.3). On the other hand, however, when # is finite these three boundedness properties are equivalent if OM - o and the following curvature-dimension condition holds: 1 ]lSess/[[~s + Ric(Vf, V f) - (VvfZ, V f} > / - K z I V f [ 2 + - ( L f) 2 (5.7.21) n for some n, K z ~> 0 and all f c C ~ (M). This is a direct consequence of Theorem 5.7.1 (1), the concentration results implied by the log-Sobolev inequality, and the following result on compactness of manifolds.
Assume that OM = 0 and (5.7.21) holds and let # be finite. If there exists r > 0 such that
T h e o r e m 5.7.8
/M
{ pn(l+c)exp I1~ ~ ( 1
max
+ C)(V/2 + 1)p ]} d# < oc,
(5.7.22)
then M is compact. Proof.
Let f e CD(M) be nonnegative with # ( f ) = 1. By (5.7.21), we have the following parabolic Harnack inequality due to [16]: t
4s
exp
+
(v/2-1)px(y) Pt+~f(Y), (5.7.23)
for all s, t > 0 and all x, y C M, where Pz denotes the distance function from x. For fixed x, letting s - (p(x)+ 1)/x/-n--K if K z > 0; s - (p(x)+ 1) 2 if K z = 0, we obtain from (5.7.23) that
1 -- P(Pt+~f) t +
(o,1)
(1 - v/2)px
oxp [-
48
(dy)
>~ct-~/2Ptf (x)min { (p(x) + 1) -n, (p(x) + 1)-n/2
•
2
P(x)
for some c > 0 and all t e (0, 1]. By this and (5.7.22), there exists C(a) > 0 such that ]]Pt[[1--.l+~ <~ C(c)t -n/2 for all t e (0, 1]. Then for any ~' E (0,~), we have [[Pt[II+c,~l+e < [IPt[[1--,l+e < C(a)t-n~2, t E (0, 1].
262
Chapter 5
By applying (5.7.1) to p obtain
Log-Sobolev Inequalities and Semigroup Properties 1 + e' and q - 1 + e, and then letting e' --~ 0, we
#(f2 log f2) ~< 2(exp[2Kt] - 1)e 12 1+ e Kze #([Vf ) + e log[C(e)t -n/2] for all t e (0, 1] and all f e C ~ ( M ) . This implies (5.1.6) with/3(r) - A log[1 + r -n/2] -nt- )~ for some A > 1. Therefore, by Corollary 5.3.6, M is compact. V1 5.7.2
Ultraboundedness
by
perturbations
Let V and W be two locally Lipschitz continuous functions, and let d# exp[V]dx, d~ - exp[W]dx. Assume that the following inequality holds for with some decreasing positive function/3 E C(0, c~)" ~(f2 log f2) ~< ru([Vfl 2) +/3(r),
f e C ~ ( M ) , u(f 2) = 1, r > 0. (5.7.24)
When OM ~ 0, this is also equivalent to the inequality for all f E C ~ ( M ) satisfying Neumann boundary conditions. Actually, for any f E C ~ (M), there exists a sequence of smooth functions with Neumann boundary conditions which converges to f in the Sobolev norm, see e.g. the proof of [186, Lemma 2.3 ]. T h e o r e m 5.7.9 Let # and ~ be as in above. Assume that (5.7.24) holds. Let ~ E C(M) be such that Ae y >>.eYrl in the distribution sense, i.e., e y A f d x >>./ f r l d # ,
f >>.0, f E C ~ ( M ) ,
(5.7.25)
where f satisfies Neumann boundary conditions if OM ~ 0. Finally, assume that there exists a decreasing/31 E C(0, oc) such that
/~1(r) ~ eE[0,1]infess~ suPM{W-v+
+ IV(V-W)12+
4- ~) ([VWI 2-27/) }.
Then we have
#(f2 log f2) ~< r
(iVfl2)+ (~ +/~1)(r/2),
r > 0, f e @(oz), # ( f 2 ) _ 1.
(5.7.26) Proof. For any f E C ~ ( M ) with #(f2) _ i which satisfies Neumann boundary condition when OM ~ O, applying (5.7.24) to f e x p [ ( V - W)/2], we have
#(f2 log f2) ~< # ( f 2 ( W _ V)) + r~(lV(f exp[(V - W)/2])[ 2) +/3(r). (5.7.27) It is easy to see that ~(IV(f e x p [ ( Y - W)/2])I 2) ~< 2t~([Vf[ 2) + l # ( f 2 l V ( Y - W)12).
(5.7.2s)
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
263
On the other hand, by (5.7.25) and using integration by parts, u(lV(f exp[(V - W)/2])l 2) 1 f2 I v ( v - w) [2)+ 5lu((Vf2 ' V e x p [ V - W])) = ,(Ivfl 2) + ~.( --
(5.7.29) 1 1 ~(Ivfl 2) + ~1 ( f2 [ v ( v - w)[ 2) - ~#(Af 2) - ~#((Vf 2, VW))
1 f2 [[V(V - W) 12 - 2r/+ [VWI2]). < 2#(IV/[ 2) + ~#( By combining (5.7.27) with (5.7.28) and (5.7.29), we obtain , ( f 2 log f2) <~ 2rp(iV fl2) + (Z + ~l)(r),
r > O.
This is equivalent to (5.7.26). Corollary 5.7.10
In the situation of Theorem 5.Z9, let Pt be the symmetric Markov semigroup on L2(#) generated by L - A + V V (with Neumann boundary condition whenever OM ~ ~). If there exists a strictly positive ~/c C[0, cx~) such that
c "-
T
dr < c~,
s > 0,
(5.7.30)
then [[Ptl]l--,~ ~< expire(t)] with
2 f cc ~ l { ~ - } - ~ l } ( t 2 ( j Pro@
For any t > 0, we have t
--
)
r-
(0,
(5.7.31/
t cT(r ) dr. By (5.7.26) and
1
Theorem 5.1.7, we obtain IlPtll2-~ <~ exp
~-~{~ -~-/31}(t~/(r)/c)dr ,
The proof is completed by the fact that
t > O.
llPtlll~oc--]lPt/21[ 22~c~"
Corollary 5.7.11
Assume that the Ricci curvature is bounded below by - K z for some K z >~ O, namely, Ric(X,X) >~ - K z I X I 2 , X E TM. Let c > v / K z ( d - 1), where d is the dimension of M, and let p be the Riemannian distance function from a fixed point o E M \ OM. If there exists a strictly positive ~/ E C[0, c~) such that
~ ' ~ [~(r)+r 131(s'y(r))ldr 2 <
,
s > O,
(5.7.32)
264
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
IVVl
where /~l(r) -- s u p { c p - V + ~[c 2 + (c +
- 2~]},r > 0, then Pt is
ultrabounded. Especially, if V = - a p ~ ( a > O, 5 > 2), then IIPtlll-~oo
~<exp
[)~1 -F/~2t -5/(5-2)]
(5.7.33)
for some )~1, A2 > 0 and all t > O. Proof. Let W = cp and u(dx) = exp[W]dx. By Proposition 3.4.8 we have the following Nash inequality
~(f2) ~< c{v(iVfl 2) + ~(f2)}d/(d+2),
r'(Ifl)- 1
for some c > 0. Equivalently (see for instance [12]), there exist Cl, c2 > 0 such that d u ( f 2 log f2) < 2 log [ClU(IVf121 + c2], u(f2) _ 1. Then (5.7.24) holds with
d cld r s } ~ ~log ( ~ r + c2),
/~(r) - sup { d 10g(cls + c 2 ) s)0
Let ~(r) - 7 ( r ) V (5.7.32) implies
r -1
r >0.
9 Since/3 and ~1 are decreasing, it is easy to see that
f2 ~ [~(r) r
+ (/3 + ~l)(SS/(r))ld
r2
J
< oc
s> ~
0 9
Then Pt is ultrabounded by Corollary 5.7.10. Next, in the case that V - - a p ~ ( a > 0, 5 > 2), by the argument in the Appendix in [223], we have (5.7.25) with zl = a252p 2(~-1) - ac3(1 + p~-l) for some c3 > 0 since the Ricci curvature is bounded below. Then we may take
~1 (r) -- sup M
{
r OL252fl2(5-1) -F c4 "+- 2ap ~ } ~ c5 + c6 r-5/(5-2) --4
6-2 for some C4, C5, C6 > 0 and all r E (0, 1]. Taking -y(r) = r -~, s = ~ . 26 Corollary 5.7.10, []Pt[[l---~eo ~ expire(t)] with
dr re(t) - 2 f2 ~ (z + Dl)(cTt'y(r)) r2
By
~ c8 + cgt -~/(~-2)
for some c7, c8, c9 > 0 and all t E (0, 1/c7]. This proves (5.7.33) for some Al, A2 > 0 . E]
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
265
C o r o l l a r y 5.7.12 In the situation of Corollary 5.7.11, assume that V is locally Lipschitz continuous and Cl-smooth in an open set whose complement has volume O. When OM ~ 0, assume in addition (N, VV}]0M >~ O, where N is the (inward) normal vector field of OM. If there exist C, a, 5 > 0 and E (0, 1/2) such that A V >f - C ( 1 + IV[) - a]VVI 2 in the distribution sense, 1 + IV[ ~ ap and 1 + IVV[ 2 ~> aIV[ 1+5. Then there exist Cl, c2 > 0 such that [IYt[[1---,c~ ~ Cl
exp[c2t-1/5],
t > 0.
(5.7.34)
Pro@ By Lemma 5.7.13 below, we may take r / - (1 - r 2 - C(1 + IV[) in Theorem 5.7.9. Let W - cp as in the proof of Corollary 5.7.11, we obtain Zl (r) -- sup M
{ cp-
V + ~(c 2 + (c + IVVI
)+
(1 + Iv1) - (1-2
Ivvl
< Cl -}- sup { c l l V l - r(1 - 2a)lVVl//8} ~< cl + cir -1/a
for some Cl,C2 > 0 and all r E (0,1], where we have used the condition 1 + IvI 1 + IvvI 2 >/~lvI 1+~, The proof is completed by Corollary 5.7.10 with ~(r) - r -~/2. K] By a simple perturbation argument of the inequality, we see that (5.7.34) remains true (with possibly different Cl, c2) if V in Corollary 5.7.12 is perturbed by a bounded function. L e m m a 5.7.13 Assume that V is locally Lipschitz continuous and C 1smooth in an open set whose complement has volume O, and (N, VV}[oM ) 0 whenever OM ~ 0. If A V ) U in the distribution sense for some locally bounded measurable function U, then for any F E C2(R) with F t ) O, A F ( V ) ~ F ' ( V ) U + F " ( V ) I V V [ 2 in the distribution sense.
Proof. Let f c C ~ ( M ) with f ) 0 and satisfying Neumann boundary conditions whenever OM ~ 0. We intend to prove / F ( V ) A f d x >1 f f [ F ' ( V ) a + F"(V)[VV]2]dx.
(5.7.35)
Indeed, it suffices to show that A F ( V ) >i F ' ( V ) U + F"(V)]VVI 2 in the distribution sense on any convex geodesic ball. Hence, we may assume that supp f C B for some convex open geodesic ball B. Now, let p B be the semigroup of the reflecting diffusion process generated by A on/~, and thB (x, y) its kernel with respect to dx (i.e. the Neumann heat kernel on /~). Let Vt = p B V on /~. We have lit E C2(/~) for t > 0, and as t ~ co, lit --~ V uniformly on B. For any x E B \ O M and v c TxM,
266
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
let {vt E TztM} with v0 - v denote the derivative process of the reflecting process {xt} generated by A o n / ) with x0 - x (see e.g. [191]). By the proof of Theorem 4.1 in [191] (see also [84], [103] for the case without boundary), we have Ivtl <~exp[ct]lv I for some c > 0 (independent of x and t), and for any g e el(B), (V, VpBg(x)) -- E(vt, Vg(xt)) and (vt, Vg(xt)) ---+ (v, Vg(x)) as t --+ 0. Since ]VV I is bounded on B and V is Cl-smooth outside a zero-volume set, we have (v, VVt(x)) - E(vt, VY(xt)). Then {IVVti}tc[0,1] is uniformly bounded and IVVtl --+ IVVI as t --+ 0. Moreover, noting that QB satisfies 0 Neumann boundary conditions on OS and AxpB -- AyQB -- ~-~pB we obtain from the integration by part formula that (recall that NVioM >~O)
" i.
i,.,o,<
where A(dy) denotes the ( d - 1)-dimensional measure induced by the volume element. Then lim AVt ~> U uniformly on supp f since dist(supp f, OB\OM) > t-+0
0. Noting that supp f C B and f, Vt satisfy Neumann boundary conditions on B ~ OM, we obtain
/ F ( V ) A f d x - lim fsu t---+0
ppf
- lim [
t---*0 Jsuppf
F(Vt)Afdx f [F'(Vt)AVt + F"(Vt) ]VVt 12]dx
1> f f [F' (V)U + F" (Y)iVY[ 2] dx. [:]
This proves (5.7.35).
Finally, we obtain in the next result the classical Sobolev inequality for a class of infinite measures. Note that by Corollary 5.3.6, this inequality does d# not hold for finite # with ~xx > 0 since M is noncompact and hence the distance function is unbounded.
Corollary 5.7.14 ro > 0 such that sup
In the situation of Corollary 3.3, assume that there exists
inf e s s z s u pf[ c p - V +
rC(0,r0] ~E[0,1]
M
+
IVVI 2
4
2
r/} < co. (5.7.36)
Then there exist Cl, c2 > 0 such that d
#(f2 log f2) ~< 2 log (Cl#(IVfl
2)
-t- c2)
,
# ( f 2 ) _ 1.
(5.7.37)
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
267
Equivalently, (5.1.6) holds with/3(r) = ~d log(c3r_ 1 + c4) for some c3, c4 > 0. In particular, (5.7.36) holds for V = ap6 with ~ > 0 and 5 E (1, 2]. Proof.
Let W = cp. By the proofs of Theorem 5.7.9 and Corollary 5.7.11,
(5.7.36) implies (5.1.6) with /3(r) -
~dlog(c3r-1 + c4) for some c3, c4 > 0
(we note that it suffices to show the inequality for small r), and (5.1.6) with this /3 can be deduced from (5.7.37). Next, we obtain (5.7.37) by taking r - (#(IVfl 2) + 1) -1 in (5.1.6) for this/3. Finally, for V - ap 5 with a > 0, 5 E (1,2], we prove (5.7.36) for small r0 by taking ~ = 1 in the right-hand side. Note that for this choice of s, the existence of ~ is not necessary.
5.7.3
Isoperimetric inequalities
For any r e (0, 1/2], let -
Theorem
5.7.15
inf #o(OA \ OM) u(A)~<~ # ( A ) v / - log #(A)
(1) If k(r) > 0 for some r e (0, 1/2], then )
# (fV2)v/log(f V 2)
21ogr -1 + 1 <~ 2k(r)log r - 1 # ( I V f [ ) + r - 1 V/l~ r - l ' f >~ O, # ( f ) - 1. (5.7.3s)
Consequently, if k(O) "- lira k(r) - ~ then r--+O
r > 0 , f ~> 0, # ( f ) = 1 (5.7.39) for a(r) "-- S ( r ) v / l o g S ( r ) , where S(r) "- inf{s >~ 2"
2 log s + 1 2k(s-1)logs
(2) Conversely, if # ( ( f V 2)v/log(f V 2)) ~< s (IVfl) + c, for some s > 0 and C >~ O, then k(r) >1 1 ( 1 -
f >t 0, # ( f ) = 1 C / v / l o g r-l).
(5.7.40)
Consequently, if
8
(5.7.39) holds for some decreasing c~, then for any e e (0, 1), k(r) >~ sup 1 - c~(s)/v/log s>0 s
?.-1
)
1 -- C
~ - l ( s v / l o g r-1 )
,
(5.7.41)
which goes to c~ as r ~ O, where oL-l(8) -- inf{t > 0, a(t) <~ s} and inf g "(Do.
268
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Proof. Let f E C ~ ( M ) with f ) 0 and #(f) - 1. By the coarea formula, for any r E (0, 1/2] we have
(IVfl) = -
f0
~ o ( { f - t} \ OM) dt >~ k(r)
k(r)
fM
d#
f yvr- ~
,It--1
--1
# ( f > t) v/log t dt
v/log t dt.
Noting that
1 ( log + 2x/log t ~< 1 +
dt for t
>
1) 2 log r-1
v/log t
r -1, we obtain
#(IVfl) />
2 k ( r ) l o g r -1
2k(r) logr -1
? l--~gr---~ -F 1 # ( ( f V 2 ) v / l o g ( f V 2 ) ) - 2 log r_l _[_1 ( r - l v / l o g r - 1 ) .
Therefore (5.7.38) holds. Next, assume that (5.7.40) holds. For any smooth domain A with 0 < #(A) ~< r E (0, 1/2], let f = 1A/p(A). We obtain the desired lower bound of k(r) by noting that #(IVfl) ~< #o(OA \ O M ) / # ( A ) . [:3 T h e o r e m 5.7.16
(1) There exist two constants Cl, C2 > 0 such that if k(r) > 0 for some r E (0, 1/2] then (5.1.1) holds for C1 - clk(r) 2 and C2 c2r -2 log r -1. If k(O) - c~, then for any e E (0, 1) there exists c(~) > 0 such that (5.1.6) holds for #(r) = c(e)S(er) 2 log S(er) + c(e), where S(r) is defined in Theorem 5. 7.15. (2) Assume that # ( d x ) - eYdx for some V E C2(M) such that
e IlVPtfll~ <~ ~llfllcc,
t E (0, 1],
(5.7.42)
for some c > O, where Pt is the semigroup of the (reflecting) diffusion process generated by L = A + V V . There exists a constant c > 0 such that if (5.1.1) holds for C2 = 0 then k(1/2) >/c/v/-C~. Moreover, if (5.1.6) holds then for any e E (0, 1/16) there exist Ce > 0 and re E (0, 1/2] such that
k(r)/> V/fl_ 1 (g" log r -1)' where ~-l(s) = inf{t > 0 9
r e (0, re],
(5.7.43)
~< s}. In particular, (5.1.6) implies k(O) - c~.
5.7
Criteria of hypercontractivity, superboundedness and ultraboundedness
Proof.
269
The first assertion follows from (5.7.39) by taking f = (g2 V 2)v/log(g 2 V 2 ) / # ( ( g 2 V 2)v/log(g 2 V 2))
for
g e C~(M) with #(g2) _ 1 and some simple calculations. Next, by Theorem 5.1.4, (5.1.6) implies
Pt -- 1 + exp[-4t/r].
IIPtllp~2 < exp [4~(r)(1/pt- 1/2)],
(5.7.44)
By (2.5.6) which follows from (5.7.42), (5.7.42) implies that
2cv~#(lVfb) >t # ( I f -
Ptfl).
By taking f = 1A for #(A) ~< 1/2 we obtain
2cv/-i#o(OA \ OM) >~2#(A) - 2#(IAPtlA) = 2#(A) - 2#((Pt/21A)2). Combining this with (5.7.44), we obtain
ClX/~#o(OA \ OM) >~#(A){1 - #(A) (2-pt)/pt exp [8fl(r)(2 - p t ) / ( 2 p t ) ] }
(5.7.45) for some Cl > 0, any t E [0, 1] and any connected smooth domain A with #(A) < 1/2. If (5.1.1) holds for (72 = 0, we may take r = C1 and fl(r) = 0 such that inf v/-iP~ \ OM) 1
t>o
p(A)(I
-
#(A)(2-pt)/p t) >~ -Cl1"
Then we obtain the desired lower bound of k(1/2) by taking t > 0 such that #(A) (2-pt)/pt - 2/3, which implies that t ~< c / l o g # ( A ) -1 for some constant c > 0. If (5.1.6) holds, then let r -- fl-1 (~ log # ( A ) - 1 ) ,
t--
r log 2 log #(A) -1 - 16/3(r)
--
r
Thus, there exists r0 c (0, 1/2] such that t ~< ~ <~ 1 if
r log 2 (1 - 16~)log #(A) -1"
#(A) < ro (note that
~(0) - cc so that r --, 0 as #(A) ~ 0). On the other hand, we have 2 - p t
2pt
for
<~--2t and 2 - p t r Pt
>~_t i f t ~ r
r
~. Then,
#(A) ~ r0, 1
p(A) (2-pt)/pt exp [8/3(r)(2 - p t ) / ( 2 p t ) ] <~#(A) t/~ exp[16~(r)t/r] - -~. By combining this with (5.7.45), we prove the second assertion.
[Z]
270 5.7.4
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
Some examples
Below we present three examples to illustrate the above results, where M is a d-dimensional noncompact connected complete Riemannian manifold, and p is the Riemannian distance function from a fixed point o. E x a m p l e 5.7.1 Assume that o is a pole and let L = A + Z with Z := VV + Z1, where V - _ap5 for some a > 0, 5 > 2, and Z1 is a smooth vector field such that lim { IZ11fl 1-5 -+- 11~7Z1lift2-5 } - 0,
p-+oo
where for each x E M, ll~TZlll :-- sup{l~TyZll : Y E T x M , I Y I - - 1}. Assume that M has nonpositive sectional curvatures and there exist Cl > 0, c2 E (0, 6 5 ( 5 - 1)) such that Ri%/> - C l - c2pa-2(x),x c M. Then Pt has a unique invariant probability measure # and is ultrabounded with
IlPtll2~ <<.exp[ct ~/(2-~)] for some c > 0 and all t E (0, 1]. Indeed, by the Hessian comparison theorem, - H e s s v >~ c~5(5- 1)p ~-2. Therefore (2.1.12) holds. Let k(r) - V/(Cl + c2r~-2)/(d- 1) (when d - 1 we set k - 0). Then Ric~> - k ( p ) 2 ( d - 1) on Bo(p). By the Laplacian comparison theorem (see w we have Ap 2 ~< 2 + 2 ( d - 1)pk(p)coth(pk(p)). Therefore, there exist c3, c4 > 0 such that Lp 2 <<.c3 - c4p 5. According to the proof of [26, Corollary 3.6], Pt possesses a unique invariant probability measure #. Then the desired assertion follows from Corollary 5.7.5. E x a m p l e 5.7.2 Assume that the Ricci curvature is bounded below. Let Z = VV for V in Example 5.7.1. By Corollary 5.7.11 and Corollary 5.3.6, there exist )~1,)~2 > 0 such that ]lPtlll__+oo ~ exp[A1 + A2t -x] if and only if
>
1).
Next, if Z = VV with V = - e x p [ a p ] for some a > 0, we have the above estimate of Iigti]l__+oo for )~ = 1. Actually, we may replace p by h(p) for some h E CC~[0, oc) satisfying h'(0) = h(0) = 0, h' ~> 0 and h(r) = r for r >~ 0. Then Ah(p) is locally bounded at o. Let V - exp[-ah(p)]. It is easy to see that A exp[V] ~> exp[V]r/in the distribution sense with ~ - Cl e x p [ 2 a p ] - c2 for some Cl, c2 > 0. Let /3 be as in the proof of Corollary 5.7.11 and /~1 be defined as in Theorem 5.7.9 with the above r/ and W = cp for big enough c > 0. Then ~1 ~< c3(1 + r -1) for some c3 > 0 and (5.7.26) holds with d# replaced by dp - exp[V]dx. Noting that ] I V - VI]~ < c~, we see that there exists c4 > 0 such that #(f2 log f2) <
r (ivfl2)+ c4(1 +
r-l),
r ) O.
5.8
Strong ergodicity and log-Sobolev inequality
271
By taking 7(r) = r -1/2, Corollary 5.7.10 implies [IPtlll__,~ < exp[m(t)] with re(t) -
-j
1 ) d r ~< )~1 + A2t -1
+ tr3/2
for some c5, )~1, z~2 > 0. Finally, the following example shows that ultraboundedness is strictly stronger than superboundedness, see [115] for the case that M - I~d. E x a m p l e 5.7.3 Assume that the Ricci curvature is bounded below and the sectional curvatures are non-positive. Let o E M be a pole and let Pt be the diffusion semigroup generated by L - A + VV + Z1, where V - - a p 2 [log(p2 + e)] 5 for some a, 5 > 0, and Z1 is a smooth vector field such that [Z1] + pl]V. ZI[] ~< c(1 +p) for some constant c > 0. Similar to Example 5.7.1 one concludes that (2.1.12) holds and Pt has a unique invariant probability measure #. By Theorem 5.7.3 (2), Pt is superbounded. On the other hand, we claim that Pt is ultrabounded if and only if 5 > 1. Actually, it is easy to check that (5.7.9) holds for v(r) - clr[log(r + e)] ~ for some Cl > 0. Then the sufficiency follows from Corollary 2.5. Next, for 5 ~< 1 there exists c2 > 0 such that Lp 2 >~ -c2p 2 log(p 2 + e). By Corollary 5.7.6 with 15-- p2 + e, we see that Pt is not ultrabounded.
5.8
Strong
ergodicity
and log-Sobolev
inequality
Let M be a connected, complete, noncompact Riemannian manifold with boundary OM either convex or empty. Consider the operator L := A + Z, where Z is a Cl-vector field such that (2.1.12) holds. Let Pt be the corresponding diffusion semigroup. We assume that Pt has an (hence unique) invariant probability measure p (see [26] for a sufficient condition of the existence and the uniqueness). Our main purpose is to compare the Ll-convergence and the hypercontractivity of Pt. Let us first explain that both properties are stronger than the L2-exponential convergence of Pt. If Pt converges in LI(#), i.e. there is a positive function { on [0, c~) with ~(t) --~ 0 as t -~ c~, such that IIPt ~111--~1~ ~(t), t ~ 0. Then, by the semigroup property, Pt converges in L 1(#) exponentially fast, i.e. there exist c,/k > 0 such that -
liFt - #[11-~1 ~ ce -~t,
t/> o.
(5.8.1)
Since ]lPt- # [ l ~ o c ~< 2, by (5.8.1) and Riesz-Thorin's interpolation theorem one has
IIP - bl2 2
t/> o.
(5.s.2)
272
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
T h e o r e m 5.8.1 (1) If (2.1.12) holds, then (5.8.1) implies the log-Sobolev inequality: there exists C > 0 such that #(f2 log f2) ~< C#(iVfl2),
# ( f 2 ) _ 1.
(5.8.3)
Consequently, the L 1-convergence implies the hypercontractivity of Pt, i.e. for any t > 0 there exists Pt > 2 such that IIPtll2__,pt ~ 1. (2) If either (2.1.12) holds or Pt is symmetric, then the ultraboundedness (i.e. [[Ptl[l~oc < oc for some t > O) implies (5.8.1) for some c,A > 0. R e m a r k 5.8.1 When Pt is symmetric, the Ll-convergence is equivalent to the strong ergodicity of Pt: lira
sup
t--.OvETP(M)
II 'P,
- ~llv~r = 0,
where P ( M ) is the set of all probability measures on M, vPt E P ( M ) is defined by (vPt)(A) := v(PtlA) for a measurable set A, and I1" [[va~ is the total variation norm defined by [[r := SUPA r infA r for a set function r In fact, if v is absolutely continuous with respect to # then
1
-~[[b'Pt--#[[var-
/M (Ptdv1)+ ~
1
dv
Since Pt(t > 0) has a transition density and # is equivalent to the volume measure (see e.g. [26, Theorem 1.1(ii)]), vP1 is absolutely continuous with respect to #. Thus, for any t > 1 one has
lIP,- # l [ 1 ~ 1 =
sup
f>/O,/z(f)=l sup
uEP(M)
IIP -I
(IP, f - 1l) ~
sup II P*- ~llv~r
vETV(M)
[[(vP1)Pt_i -#[[var -
sup
uETP(M)
#
(IP t - i d(/]PI) 11) d#
-
Therefore, according to Theorem 5.8.1(1), (2.1.12) the strong ergodicity of Pt implies the log-Sobolev inequality. To prove Theorem 5.8.1(1), we need the following interpolation theorem due to Peetre [152]. A proof of such a result [101, Theorem A.1] has been presented in [101]. In the following version we figure out the explicit relationship between the interpolation constant and the original ones. T h e o r e m 5.8.2 (Peetre's interpolation t h e o r e m ) L e t r162162 be three nonnegative increasing functions defined on [0, c~) such that r = r162
r--,
9
.
~
9
~
~
~
~~,~
9
II
I ~
~.~
V
~
>'
~
II
I~
~
~
~-.
o ~
~
~ ~ ~ i~+ ~
~
~.
I
I
~.>~
+
P
~-e~ ~
o
--.
~
$~~-e-
~,
~
~
~
o-, +
o 0-q
I
/A
~'~
~
+
!~
I ~ ~ ~::: q,~, ~__,
o~
I ~--.~
~..
~ ~
~
~.~.
~
~
~ . .~. I
<1 ~
V
~
o
~
,-,,
+
M
~G
/A
0
~o
~
i~
~
0
~
I~
~o
9
~ ~:
~
~
~
rl
c~ .
~ ~
~
9
~
Oq
-.
o
(~
.
~
c~ ~
<
/A
/A
~
~
-~-
~
~
o
"1::
'-"-'
-~
~.
~ , ~ " o'-", _1_~,.~ ...1_~..~ ~ ,-~
/A
bo
a~
9
/A
~
/A
8
(~I~
~
~'-
~~
~
~
8
O
~
w
c~
~
BL~ -.
9
\V
['O
Jr-
~
,-"-,
+~
~-~
b,
I~
~
w
~
m q~
-~,
<
o
~.~
C~
~ ~
~
~o
O-~
~-
~.
~
,..
~
~
~
~]
~
~
V
~
II ~ ~
V
8 ~~-
A
~_~S.
I
.~.\V
~
u 0
"'-,1
-"
o
O~
_
o
0~
274
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
for some c2, A2 > 0 and all t ~> 0. Therefore, there exists C3 > 0 such that
#(Ptf 2 log Ptf 2) <~c3e -~2t#(f2 log f2) + c3,
t ) 0, #(f2) _ 1.
Combining this with (5.8.7) for a proper choice of t > 0, we obtain
(f2 log f2) < A
(ivf]2)+ B,
(5.s.s)
(f2) = 1
for some A, B > 0, which implies the super Poincar6 inequality. Thus, as explained in the proof of Theorem 5.7.3, (5.8.8) holds for some A > 0 and
B=O. (b) By Theorems 5.1.4 and 5.7.1, if Pt is ultrabounded then (5.8.8) holds for some constants A, B > 0. Thus, the Poinca% inequality holds so that IIPt- ~112---+2 ~ e-t/C holds for some C > 0 and all t ~> 0. Therefore, if IIPto - ~111-+2 < cx:) then for all t > 0 one has
[[Pt+to
--
]-t[[1---~l ~< [[Pt0
-
-
#[]l~2[[Pt
-
-
#[[2-~2 ~ e-t/C[[Pto #[[1---+2.
[-]
To show that the L 1-convergence and the hypercontractivity is incomparable, let us first introduce a criterion on the strong ergodicity which is equivalent to the L 1-convergence for the symmetric case according to Remark 5.8.1. d2
d
C1
Consider L "- a(x)-~--fix2 + b(x)-d--~x, where a, b E ([0, co)) x b(r) with a(x) > 0 for all x >~ O. Let C(x) = a(r) dr' x e ]R. Then the
Lemma
5.8.3
fo
corresponding reflecting diffusion semigroup Pt is strongly ergodic if and only
if 5 "-- ~o~ e - C(x)dx ~ c~ ~eC(r) dr
< oc.
(5.8.9)
Proof.
According to [78], the strong ergodicity is equivalent to supx>0 E~T0 < oc. So, it suffices to show that the later is equivalent to (5.8.9). Let TO := inf{t ~> O:xt = 0}, where xt is the reflecting L-diffusion process. Let
F(x) "-
e -c(~)dr
~ds,
a(s)
x ) O.
We have LF(x) = - 1 and hence for x > O, 0
< E~F(xr
-
F(x) - E~(To A t),
t>0.
Letting t --+ oc we obtain ExT0 ~< F(x) and hence (5.8.9) implies supx>0 ExT0 < c~. On the other hand, letting Tn "-- inf{t >f O ' x t >f n} we have
F(x) - EXTo A Tn + EXF(To A Tn) <<.F(n)~X(Tn < 70) + ExTO,
n>x. (5.8.10)
5.8
Strong ergodicity and log-Sobolev inequality
Since for G ( x ) " -
fo x e-C(~)dr
one has L G -
O, it follows that
G(n)IpX(Tn <
F(x) - EXG(xroAr,~)-
275
7"0),
n > x.
Combining this with (5.8.10) we arrive at F ( x ) <~ Ex~-0 +
F(n)a(x) G(n) '
(5.8.11)
n > x.
This implies that F ( c c ) < oc provided SUPx>oEZTo < oc. Indeed, since [ ~ eC(~) J0 a ~ - ) d r < oc, if F(oc) - oc then G(oc) - oc and F ( n ) / G ( n ) ~ 0 as n --, co. Thus, by letting n ~ oc, we obtain from (5.8.11) that F ( x ) <. EZTo for all x > 0 and hence SUpx>0 EX~-0 = oc. d2 Consider the Ornstein-Uhlenbeck operator L := dx 2
E x a m p l e 5.8.1
d
~X~dx
on [0, co). It is well-known that the semigroup Pt of the reflecting L-diffusion process is hypercontractive. But according to Theorem 5.8.3, Pt is not strongly ergodic since (5.8.9) does not hold. Therefore, Pt does not converge in the L 1norm by Remark 5.8.1. d2 d E x a m p l e 5.8.2 Let M - [0, c~) and consider L " - d x 2 + b(x)-~z , where b(x) . -
~'(x) Z(x)
1 ~(x)'
x/> 0
with ~/constructed as follows. For any n ) 1, let Cn E Vet[0, oc) be nonnegative such t h a t Cn][n,n+e-n]c --0 and
d/)nl[n+e-n/4,n+3e-n/4] Set 7(r) -- (1 + r) -2 + E
max Cn -- en( 1 + n) -2.
Cn (r), r ) 0. Then Pt is strongly ergodic and hence
n)l
L 1-convergent but not hypercontractive. Proof.
We have C ( x ) " - fo x b ( r ) d r - - log ~(x) - fo x ~dr (r)'
x/> 0.
Then
dr I , exp[-C(x)] - ~(x) exp [ fo x ~(~)
exp[C(x)] -
d
ax ~xp
dr] [/0 x ~(~)
276
Chapter 5
Log-SobolevInequalities and Semigroup Properties
oO Therefore, ~0 eC(x)dx = 1 and f0 cr e -C(~) dx 1 ~
eC(y)dy =
/0
~(x)dx ~
1 + r -F /0 --------~ dr fi 1 +1n 2
By Lemma 5.8.3, Pt is strongly ergodic. On the other hand, observe that
l (l + x)3 >/ j~oX7(r) dr >~ g(l 1 + x)3 - E
fn+e-n(I + r)2dr
n>~l~n
1 1 ~> 5(1 + x) 3 - ~-'~(2 + n ) 2 e - n - " 5(1 + x) 3 - el.
n~l
Then for #([0, x])"- ~0 x eC(r)dr,
e-C(z)#([O,z])dx=
7(x)
e-Ca ~0~ e(l+x)a/3 (1 + x) 2 dx - ~0ec7(x)dx since
I(n)
exp
[/0
dx
- c~
~,(x)dx < ee. Moreover,
._ [f\n+e- 0 e-C(y)dY)(Z+e-neC(y)dY)(C~ log f~+e-~ eC(y)dy
:(fJ0 n-l-e-nV(y)exp [j~0y v(r) dr] Y/ [ fn+e-ndr)](j~on-+'e-n d exp-j0 v-(r ~V~)) ~ c 2 ( f n+3e-n/4 an+e-~/4
en
(1 + n) 2 exp[(1 +
y)3/3]dy
)
e-(l+n)3/3(1 + n) 3
/> c3(1 + n) - ca
for some c2, c3, c4 > 0. Thus lim I(n) = oc. Therefore, according to Theorem n----~(:x3 5.5.1, the log-Sobolev inequality does not hold. [5 5.9
Notes
The hypercontractivity was int1'oduced by [148] for the Mehler (or OrnsteinUhlenbeck) semigroup, and then described by [98] using log-Sobolev inequalities. The super log-Sobolev inequalities were used in [69] to study the superboundedness and the ultraboundedness of symmetric Markov semigroups. See
5.9
Notes
277
e.g. [99], [72], [13], [68] and references within for proofs and history remarks of results included in w w is modified from [212]. It is well-known in the literature that the hypercontactivity (i.e. 5(P) = 1) for a symmetric Markov operator implies the existence of spectral gap. In the context of symmetric contraction Cosemigroup (Pt)t>~o with Ptl = 1, the Rothaus-Simon mass gap theorem says that the hypercontractivity (i.e., there exists t > 0 such that b(Pt) = 1)implies the existence of the spectral gap, see [163], [166] and also [99]. More precisely, if 5(PT) = 1 then (see e.g. [72, Lemma 6.1.5])
R1 (Pt) ~ 3-[t/T]/2,
t ~ T,
where [r] := sup{n c Z : n ~< r}. According to Theorem 4.1.4, this implies the following lower bound estimate of the spectral gap of the generator (L, ~(L)): gap(L) " - i n f { - p ( f L f ) ' f
e ~(L), #(f2) _ 1, #(f) - 0 } ~>
log 3 2T
(5.9.1)
Therefore, 5(PT) -- 1 implies RI(Pt) ~ 3-t/(2T) for all t ~ 0. Next, in 1998, [2] proved that gap(L) > 0 provided there is t > 0 such that Pt is hyperbounded and is uniformly positivity improving ([120]):
inf{#(1APtlB): #(A), #(B)/> ~} > 0,
c>O.
Obviously, the uniformly positivity improving property is stronger than the ergodicity. In 2000, [217] introduced the notion of uniform integrability for linear operators and studied the above question for ergodic, uniformly integrable, positivity-preserving operators (see [217, Problem 3.10]). Moreover, the above mentioned Aida's result was generalized or extended in [102], [94] for general positivity-preserving operators. Very recently, the essential spectral radius of Markov operators was estimated in [218]. Proposition 5.2.5 does not provide any negative answer to Wu's problem 3.10 in [217]: whether an ergodic hyperbounded Markov operator posses a spectral gap, since the operator constructed in the counterexample is not positivity-preserving. Wu ([217, Example 1.8]) also provided an example of hyperbounded (but non-C0) semigroup without spectral gap. w is based on [100] and [159], where Theorem 5.3.1 and the basic argument in this section are due to [100]. w is a continuation of w and w where some results can be found in [140] for K = 0 and p a probability measure. See [41], [43], [197], [185] for the study of general Sobolev-Nash type inequalities of jump processes.
278
Chapter 5
Log-Sobolev Inequalities and Semigroup Properties
w and w provide an application of the Hardy inequality to log-Sobolev inequalities. The log-Sobolev inequality was first connected in [22] with a Poincar~ type inequality in Orlicz norms, and a generalization of the latter was then studied in [45], [46]. The results concerning (5.1.1) for the birthdeath process and the one-dimensional diffusion processes can be found in [138], [22], [47], but those concerning (5.1.6) are new. Theorem 5.6.1 provides a quite complete description of the curvature condition (2.1.12), where the equivalence of the first seven statements can be found in [13] and references within (11) and (12) are essentially due to [156] and the others come from [211]. This theorem is crucial for the further study in w Subsections w and w are based on [52] and [195] respectively, where the Sakry-Emery criterion (Theorem 5.6.2) is initiated in [14]. w and w are reformulated from [159]. It might be useful to mention that in [200] a counterexample is constructed to show that #(exp[Ap2]) < ec is not enough to implies (5.1.1) if A < gz/4. Combining this with Theorem 5.7.3 we have 5 E [Kz/4, Kz/2], where 5 is the smallest positive constant such that under (2.1.12) the condition #(exp[Ap2]) < oc implies (5.1.1) for all A > 5. But the exact value of 5 is yet unknown. Moreover, it should be useful to find out also some equivalent statements for the dimension-curvature condition (see e.g. [13]). Finally, the results in w is due to [210].
Chapter 6 Interpolations of Poincar6 and Log-Sobolev Inequalities Let (E, ~ , #) be a probability space, and (L, ~ ( L ) ) a densely defined linear operator generating a Markov C0-semigroup Pt on L 2 (p). Assume that # is an invariant measure of Pt so that Pt is contractive in LP(p) for all 1 ~ p ~ c~. Let 8(f,g) := -#(fLg) for f, g E ~(L). One has d~(f, f) ~> 0 since Pt is contractive in L2(#). If (8, ~ ( L ) ) is closable in L2(#) then we write (d~, ~ ( 8 ) ) for its closure. Recall that the Ponica% inequality reads
#(f2) <~CS(f, f ) + #(f)2,
f e ~(L),
(6.0.1)
where C is a constant. In this chapter we study functional inequalities with so-called additivity property which interpolate (6.0.1) and the log-Sobolev inequality E n t , ( f 2 ) ._
#(f2 log[f2/p(f2)]) ~ 2 C 8 ( f , f),
f e ~(L),
(6.0.2)
where C is once again a positive constant. Consider the following inequality: sup pc[l,2)
#(f2) _
(iflp)2/p
r
< d~(f, f)
'
f e ~(L)
'
(6.0.3)
where r c C[1, 2] is a decreasing function and is strictly positive in [1,2). Obviously, when r - 1 then this inequality reduces to (6.0.1). Moreover, it is easy to check that (see [121, Lemma 1]) sup p[#(f2) _ lz(lflP)a/P] = lim pc[l,2) 2 -- p p---~2
pip(f2) _ lz(iflP)2/p] = E n t , ( f 2) 2- p
'
with r 2 C ( 2 - p)/p the inequality (6.0.3) coincides with (6.0.2). Therefore, (6.0.3) is a correct generalization of (6.0.1) and (6.0.2).
Chapter 6
280
Interpolations of Poinca% and Log-Sobolev Inequalities
According to Ledoux [125, Proposition 4.1] (see also [121, Corollary 5]), one concludes that (6.0.3) possesses the additivity property and hence applies to the infinite-dimensional case as (6.0.1) and (6.0.2) do, namely, if (6.0.3) holds for some two forms (s ~(s on (Ei, ~i, #i) respectively, i = 1, 2, then it also holds for the following form on the product probability space:
e(f ' S) "-- fE ~2(S(xl")' f(xl"))"l(d~l) -+-/E ~l(S("x2)' f("x2))"2(dx2)' 1 2 where we set s g) = c~ for a function g ~ ~(s i = 1, 2. In w we introduce more properties of this new inequality, in w we present some criteria of this inequality by connecting with the F-Sobolev inequalities and by using the Hardy's inequality. Finally, the transportation cost inequality on Riemannian manifolds is studied in w by using (6.0.3).
Some properties of (6.0.3)
6.1
P r o p o s i t i o n 6.1.1
#(f2) _
For any f e L2(#) and f "- f -
#(lflP)2/p< #(f2)+ (I -p)#(lflP)2/p,
#(f), one has p e [1, 2].
(6.1.1)
consequently, Ent~(f 2) < Ent~(f 2) § 2#(f2).
Proof.
(6.1.2)
For ~ > 0 and f e n2(#), let gt "- ( # ( f ) + t f ) 2 + ~ ,
t e [0,1].
Consider r
.- #(gt) - ~(~/2)2/p = ,(f)2
_ t2~(12) + t2~(iflP)2/p
+ ~ + t2,(iflP)2/p
_
,(~/2)2/p.
For t E [0, 1], one has
r
=2t,(111;) ~/;
r
=2#(If[P) 2/p - 2(2 -P)#(~/212(1-p)/p
-
2.(F~/~) (~-;//;,~,~t :o(;-~//~ (#(f) + tf)f), , (p-22)/#~gt
(#(f) + t f ) f ) 2
- 2(p- 1). (g~/~) (~-~)/~. "~g,(~-~)/~:~) ~2.(IflP) 2/p -
2(p- 1)#(g~t/2) (2-p)/p#(gt, (p-2)/2 f2).
On the other hand, by the Schwartz inequality,
.(IfF)
=
.(Ill p~ (p- 2)/4~ (2-;)/4)
<, ,(~/2)(2-p)12,.~2 (p-2)12)p12 ~:
gt
,
6.1
281
Some properties of (6.0.3)
it follows that ~< 2 ( 2 - p)#(l]lp) 2/p,
r
t e [0,1].
Thus, r
- r
- r
/0 1 ds /0s r
-
(r)dr ~ ( 2 - p)#([flP) 2/p
Then the proof of (6.1.1) is completed by letting e ~ 0. Finally, (6.1.2) follows 1 from (6.1.1) since 2 -1 p [#(f2)_ #(fp)2/p] __+ ~Entu(f2) as p ~ 2. Next, let us look at the semigroup property induced by (6.0.3). We shall use the following two assumptions, where the first assumption follows from the diffusion property in the sense of [13].
4--qE(f(q+l)/2,f(q+l)/2) (1 + uniformly positive f C L ~ ( p ) ~ ~ ( i ) . (A6.1.1)
s
(A6.1.2) Pt
f) -
for any q > 0 and any
is symmetric.
P r o p o s i t i o n 6.1.2
If (A6.1.1) holds then (6.0.3) is equivalent to ~< exp[-2t],
pE[1,2)
#(f2/p) _ #(f)2/p f)0,
fcL2(#),
(6.1.3)
t)0,
where and in the sequel, we assume that f is not constant so that the left-hand side makes sense. Conversely, if (A6.1.2) holds then (6.0.3) implies (6.1.3). Proof. It suffices to prove for uniformly positive f E L ~ ( # ) ~ ~(L). such a function one has d
2
- p #((Ptf)(2-p)/pLPtf) -
For
-2
p 8((Ptf)(2-P)/P, Ptf)
for t/> 0. It suffices to note that by Lemma 5.1.3 and (A6.1.2) one has
8 ( ( P t f ) (2-p)/p, Ptf) ) p(2 - p ) 8 ( ( P t f ) 1/p, (Ptf)l/P), while it is easy to see that (A6.1.1) implies the equality.
K]
Now, we consider the concentration of # using (6.0.3). This will provides a necessary condition for this functional inequality.
Chapter 6
282
Interpolations of Poinca% and Log-Sobolev Inequalities
P r o p o s i t i o n 6.1.3 Suppose that (E, ~(d~)) exists and is a Dirichlet form such that (6.0.3) holds. Let g(A) "-- sup ( ( 1 - a J t 2 r
21(2-p),
~ e [0,2/v/ar
pE[1,2)
where 2 / v / a r ) " - oo if r - 0, and a - 1 for the diffusion case and a - 2 for otherwise. For any p with L3(p) <<.1 we have #(p) < ~ and IEd#
L2(ar
In particular, if r sup eE(0,1)
Lio d#
where r
g(A)e x(p-r) dA ~ r - 1#(p)'
r > #(p).
(6.1.4)
- 0 then (1 - ~)21[2-r
<~ r -
e [1,2] 9 r
v(p)'
~< t}, t > 0.
Proof. Since (6.0.3) implies the existence of spectral gap, one has #(e xp) < oc for some A > 0. Let P n "-- P A n and Hn(A) " - #(eXP"), A > 0. Applying (6.0.3) to f "- e xp"/2, we obtain from Lemma 1.2.4 that a~ 2
H n ( A ) - Hn(pA/2) 2/p <<.---~r By an argument of [8] (cf. page 10 in [121]), we arrive at #(e ;~(pn-'(pn))) ~< 1 - - - ~ - r
,
A e [0,2(ar
Letting n -+ oo we obtain 1 g(A),
#(e),(p_,(p))) <
x
[0, 2(ar
Then the first assertion follows by noting that d#
g(A)e~(P-r)dA dO
e_),(r_~(p)) d A _
)dA
1 -
If r
g(A)# dO
v(;)
= 0 then for any ~ E (0, 1), by taking p - r
one obtains
g(A) /> ( 1 - c ) 2/(2-+-1(4~13~2)). Thus the second assertion follows.
D
6.1
283
Some properties of (6.0.3)
C o r o l l a r y 6.1.4
Assume that (6.0.3) holds. For any a E (0, 1] with ~ "-
p--.2 p)~ < oc, there exists c(a) > 0 such that lim ( 2r- (p) #(exp[c(a)t-1/(2-~)p2/(2-~)]) < co,
t >rl.
Since r / < oc, for any t > r/there exists rt E (0, t) such that
Pro@
2-
Taking r
r
(0, rt].
1/2 one has
/o
(1
-
--
e)2/(2-r
4 -(a'x2t/2)l / ~ e "x(p- r) dA
exp[A(p - r) - cotl/~A2/~]dA,
2~art where co "- (a/2) 1/~ log 4 ~< log 4. Let P0 > 0 such that Ao "-- (po/[2(log 4)tl/~]) ~/(2-c~) > (2/[art]) 1/2. For any A E [Ao, Ao + 1] one has )~(P0 -- r) -- (log 4)tl/c~)~ 2/c~ ~ )~0(P0 -- T) -- (log 4)tl/c~()~ 0 -~- 1) 2/a
=2(log 94)t 11c~"'o ~2/c~ -- rAo -- ( l o g 4 ) t l / ~ (1 + Ao) 2/~ )(log 3)t 1/c~~2/c~ "'o -- C(t, c~) - c(c~)t -1/(2-c~) Po2/(2-c~) - C(t, ct) for some C(t, a), c(a) > 0. Therefore, f0cc(1 _ ~)2/[2-r -l(4e/a~2)]e~(p-r)dA Ao+l
~> e-C(t, a)
exp[c(ct)t-1/(2-a) p2/(2-a) ]d~
= e x p [ - C ( t , a) + c(oL)t -1/(2-a) p 2 / ( 2 - a ) ] ,
P > Po.
Hence the proof is completed by Proposition 6.1.3. [3 Moreover, we present below a simple perturbation result. Let ~ ( E ) denote the class of real measurable functions on (E, o~), and let F : ~ ( F ) • ~ ( F ) ~ ( E ) be a symmetric bilinear mapping satisfying: (i) ~ ( F ) is a sub-algebra of ~ ( E ) , F ( f , f ) >t 0 for f e ~ ( F ) , 1 e ~ ( F ) .
284
Chapter 6
Interpolations of Poincar6 and Log-Sobolev Inequalities
Proposition 6.1.5 Assume (i) and suppose that 8 is realized by 8 ( f , f ) # ( F ( f , f ) ) for f e ~ ( F ) ~ ~ ( 8 ) . Let fit " - e Y d # be another probability measure with Y e L ~ (#). Let 5z(Y) "- essz sup Y - essz inf V. If (6.0.3) holds then
sup pC[l,2)
p(lflP)2/p
fz(f 2) r
<. e ~ ' ( v ) # ( F ( f , f)),
f E :~(e)N ~(F).
For any nonnegative f e ~ ( F ) ~ ~ ( 8 ) , one has
Proof.
2 _ p
~t(f) "-- f2 _ tfp +
p
>~ 0,
t > 0, p E [1 2).
Next, it is easy to see that for any probability measure #, there holds #(f2) _ f~(fp)2/p _ inf/2(~t(f)). t>0 Therefore #(F(f))
~< e a~(v) inf # ( ~ t ( f ) ) = ea~(v)#(f2) _ #(fp)2/p t>o # ( F ( f ) ) 8 ( f , f) "
V] Finally, we consider the case where V is "Lipschitzian". We assume further that (F, ~ ( F ) ) satisfies: (ii) If f, g c ~ ( F ) , then F ( f g, h) = f F(g, h) + g F ( f , h), h c ~(F). (iii) If f, g E 2 ( F ) , then for any G e CI(]~) one has G ( f ) E 2 ( F ) and F ( G ( I ) , g) = G ' ( I ) F ( I , g). 6.1.6 Assume (i), (ii) and (iii). Let 8 be realized by (F, ~ ( F ) ) . Let V e ~ ( F ) with [[F(V, Y)[[~ < oc such that for some constant c > O,
Proposition
F ( f e V/2, f e V/2) <. 2f2e VF(V, V ) + c e V F ( f , f),
f e ~(F).
(6.1.5)
If fit "- eV# is a probability measure, then (3.3.3) implies #(f2) <~ r p ( F ( f , f)) + 3(r)#(]f[) 2,
r > 0, f e ~ ( L ) ~
~(F),
where
/3(r)'--inf { -teN cr t
"
r' , Y>O, 1-/3(r')#(V < - Y ) - 2 , ' l l r ( V ,
V)ll
cr'}
i> - -
r
r>0.
Consequently, if (6.0.3) holds with r - 0 and (~, ~ ( ~ ) ) is a Dirichlet form with core ~ ( L ) (7 ~ ( F ) and 8 ( f , f) - p ( r ( f , f)) e ~ ( 8 ) N 2 ( r ) , then (6.0.3) holds for 8 and some r with r - 0 in place of 8 and r respectively. In particular, if r = c(2 - p ) a for some c > 0 and a e (0, 1], then we may take r - d ( 2 - p)a for some c' > O.
f
6.2
285
Some criteria of (6.0.3)
Proof.
For f E ~ ( L ) ~ ~ ( F ) with using (6.1.5) we obtain
~(Ifl)
#(f2) ~< c r # ( r ( f , f)) + 2tilt(V,
- 1, applying (3.3.3) to fe y/2 and
v)ll~p(f 2) +
~(r)#(lfle-V/2) 2.
Noting that ~(]fle-V/2) 2 ~ eN~(lf]) 2 + fz(lfle'V/21{V<_N)) 2 eN#([fl) 2 + #(f2)#(V < - N ) ,
N>O,
we obtain ~p(r(f
#(f2) ~< (1 -/3(r)#(v
, f ) ) + ~Np(Ifl)2 < - N ) - 2 ~ l l r ( v , v)ll~)
+
This implies the first assertion. Since (6.0.1) holds for some C > 0, it is easy to see that the weak Poinca% inequality with q h ( f ) . - I l f l l ~ holds for o~ (see [158, Theorem 6.1]). Thus, by the first assertion and Proposition 4.1.2, the form d~ also fsatisfies (6.0.1) for some C > 0. Therefore, the second assertion follows from Theorems 6.2.1, 3.3.1 and 3.3.3. Finally, if (6.0.3) holds for r - c ( 2 - p)~, then it follows from Corollary 6.2.2 that (3.3.3) holds for/3(r) - exp[cr -1/~] for some c > 0. Next, it is easy to see that L c ( V ) <<.IIF(V, V)II~. Then by the concentration result deduced from (3.3.3), there exist A1,A2 > 0 such that p(V < - N ) ~< Ale -~2N for all N > 0. Taking r ' - r/4 and N -- /~21 log(4A1/3(rt)), we conclude that /3(r) ~< exp[c'(1 + r-1/~)] for some c' > 0 and small r > 0. Since (6.0.1) holds for go the proof is now completed by Corollary 6.2.2. D E
6.2
Some
criteria
o f (6.0.3)
We first connect (6.0.3) with the F-Sobolev inequality so that known criteria of the latter and the super Poinca% inequality can be applied to the former. T h e o r e m 6.2.1
(1) Assume the F-Sobolev inequality
# ( f 2 F ( f 2 ) ) <. C18(f, f ) + C2,
f e ~ ( 8 ) , # ( f 2 ) _ 1,
(6.2.1)
where C1, C2 > 0 are constants and F is a nonnegative increasing function with F(ce) "- lim F(r) - oc. Let r---~oo
9 p(t) "- 1 V supr(2-P)/2(t- F(r)), r>O
t > 0,; e [1, 2].
286
Chapter 6
Interpolations of Poincar6 and Log-Sobolev Inequalities
If (6.0.1) holds for C - Co then (6.0.3) holds for r
"-
r
r
A Co, where
C o ( 2 - p ) + t>o inf ~1 { C1 +CoC2+-~Co(2-P)(-~t 1 ) p/(2-p) #p(t) 2/(2-p) } 1
~ C~
C1 + CoC2
p/(2-p)
tp
[1+ 2 ( 2 ) ] +
tp "-- r~linf{ 2 -p2 rF'(r)+
'
F(r)},
p E [1, 2).
(2) Assume that (6.0.3) holds for some r with r ~(t)'-- sup
1-
t -(2-p)/p
,
r
pC[1,2)
- O . Let t~>l.
If (8, ~ ( 8 ) ) exists and is a Dirichlet form, then for any 5 > 1, (6.2.1) holds
v/~+l
for C2 - O, C1 - 5(v/5 - 1) and
1/lV(,/~) -
F(r) "-- r J1 Proof. have
~(s)ds,
r>5.
(a) For f C L2(#) with # ( f ) - 0 and #(Ifl p) = 1, let 0 "- #(f2). We
sup r2-P[t-F(r2/O)] = 0 (2-p)/2 sup r(2-P)/2[t-F(r)] - 0 (2-p)/2 ~p(t), r>O
r>O
t>O.
Then f2F(f2/O) ~ f2t - 0(2-p)/2 ~p(t)lflP. Since
~(Ifl p) - I we obtain #(f2F(f2/O)) >>.t O - 0 (2-p)/2 4~p(t) >>.t ( O - 1 ) - c(t)O,
where c(t)
" - -
sup{O -p/2 ~p(t)
-
t/O} ~
0>1
=2
-
sup {r p/2 ~p(t)
-
tr}
rE(O,c~)
p (~)p/(2-p) p ~p(t)2/(2-p)
2
Thus, by (6.0.1) with C - Co and applying (6.2.1) we obtain t[#(f 2) - #(If Iv)2/v]
{ C1 -[- CoC2 -[- c0(2 - p),( ~; / ~/(2-~) 2
}
~(t) 2/(2-p) 8(f, f)
t>O.
6.2
Some criteria of (6.0.3)
287
Combining this with (6.1.1) and using (6.0.1) with C = Co once again (note that [[]]]p ~< !1][[2), we conclude that for all f e ~(oz),
#(f2)_ #(iflp)2/p <~C 0 ( 2 - p) 1{ Co(2 - p) 'f /P)P/(2-P)#(t)2/(2-P)} go(f, f) +7 C1+ CoC2 + 2 for all t > 0. This, together with (6.0.1) for C = Co, implies (6.0.3) with r = Co A r Thus, to complete the proof of (1) it remains to show that ~p(tp) <~tp. Since F >~ 0, one needs only to prove that G(r) := r(2-P)/2(t- F(r)) is decreasing in r E [1, oc). Indeed, one has - 2--f-r F'(r)}~ ~< 0, ) 2-p (b) Let A~ n >~ 0. We have
=
r >~ 1.
{(~n+l ~> f2 >1 5~} and f~ := ( f - 5n/2) + A (5(~+1)/2 -5~/2),
U(fn < r
A)+
<~r
A) + >(f~)#(f2 >I 5~)(2-p)/p f n ) + #(f2n)5-n(2-P)/P, p e [1, 2].
< r
Since (8, ~(d~)) is a Dirichlet form, we obtain (see the proof of Theorem 3.3.3) (X~
OO
E(S, f) >f ~ W(f~, f~) >1~ ~(5~)#(f~) n=O
n=O
) E ~(hn)#(f2 ) (~n+l)[(~(n+l)/2 _ (~n/212 n--0
v~ + 1
((t)"(f2 > 52t)dt n=0
= ( v / 5 - 1)5
n-1
d.
~(t)dt
v +l
(V~- 1)5
= ~ + 1 P(f2F(f2))' where F(r)"- -rl/1 (~/~)vl ~(s)ds,
r > 0.
Assume that (6.0.1) holds and let a e (0, 1]. If (8, ~(E)) is a conservative Dirichlet form then the following three statements are equivalent to each other:
Corollary 6.2.2
288
Chapter 6 (1) (2) (3) (4)
(6.2.1) holds (6.0.3) holds (3.3.3) holds Let ~ ( z ) :=
Interpolations of Poincar@ and Log-Sobolev Inequalities
for F(r) "= [log+ r] ~ and s o m e C 1 , 6 2 . for r C ( 2 - p)~ for some C > O. for/3(r) "- exp[cr -1/~] for some c > O. Ix[ log~(1 + Ix[). There exists C > 0 such that
CS(f, f),
tl(f - #(f))2ll
f e .~(8).
(6.2.2)
Proof. Since (6.0.1) holds, the equivalence of (1) and (3) have been included in Corollary 3.3.4 (1). Moreover, by repeating the proof (a) of Theorem 5.4.3 with the present ~P, we conclude that there exists a constant c > 0 such that
#(f2(log+ fa)a) _ c ~< IIf211
< #(f2(log+ f2)c~) + c,
# ( f 2 ) _ 1.
Then the equivalence of (1) and (4) follows from the Poincar@ inequality. So, it remains to check the equivalence of (1) and (2). If (6.0.3) holds for r = c ( 2 - p)~, then by Theorem 6.2.1(2) one has (6.2.1) for (72 = 0, some C1 > 0 and -rl
F(r)-
jr1(r/2)V1~(s)ds,
where ~(s):= For s > e, let p -
pC[l,2)
c(2 - p ) ~
,
s>l.
2 - (log 8) -1. W e obtain
1 -- 8 - ( l ~
(s) ~>
1 - s -(2-p)/p
sup
c(log s ) - a
1 -- e - l I p
c
1 -- e -1/2
(log s)
>/
c
(log s)a,
s ~> e.
Therefore, there exists c' > 0 such that F(r) ~ c'(log + r) ~ - 1 for all r > 0. Finally, for F ( r ) - [log+ r] ~ one has tp - { 2a
2(1 -- OL) a-1 2 -p
(2 - p ) ~
for some constant c > 0 independent of p and a. Then the proof is completed by Theorem 6.2.1 (1). The above results enable us to verify the following concrete model on Riemannian manifold, from which one can see how the inequality changes continuously from the Poinca% to the log-Sobolev.
6.2
289
Some criteria of (6.0.3)
C o r o l l a r y 6.2.3
Let E - M be a d-dimensional noncompact connected complete Riemannian manifold with Ricci curvature bounded below. Let p(x) be the Riemannian distance between x and a fixed point o. Consider #(dx) := ZeVdx, where V E C ( M ) such that V + Ap ~ is bounded for some r E (1,2] and A > O, dx stands for the Riemannian volume measure, and Z is the normalization. Let E ( f , f ) " - #(IV f[ 2) with ~ ( E ) " - H2'1(~). Then there is C > 0 such that (6.0.3) holds for
r
-
-
C(2
-
-
p)2(r-1)/r.
This r is optimal in the sense that (6.0.3) does not hold for any r satisfying
lira
r
p----~2
-p)-2(1-r)/r
(6.2.3)
__ O.
If, in particular, Ric ~ - K for some K ~ 0 with A > v / K ( d - 1), and IIV + Aprll~ <~ C' for some constant C' independent of r, then the constant C can be chosen independently of r too. Proof. By Corollary 3.4.10, we have (3.3.3) for ~ ( r ) " - exp[c(1 + r-1/~)], where a := 2 ( r - 1)/r. Then by Corollary 6.2.2, (6.0.3) holds for r := C ( 2 - p)a for some C > 0. On the other hand, if (6.0.3) holds for some r satisfying (6.2.3), then it follows from Corollary 6.1.4 that exp[Ap(x)2/(2-a)]dx < cr
A>O.
> 1, this is impossible since according to the volume comparison 2-a theorem one has (see e.g. [33]) Since
Vol(B(x, 1)) ~> e x p [ - k ( p ( x ) + 1)] for some k > 0 and all x E E, where the left-hand side stands for the volume of the unit geodesic ball around x. Finally, we prove that C can be chosen independently of r c [1, 2] provided A2 > K ( d - 1). Let h E C~(I~) with h(r) - 0 for r < O,h(r) - r for r ~> 1 and 0 <~ h ~< 1. By the perturbation result Proposition 6.1.5, we may assume that V - - A h ( p ) ~. To make use of Theorem 3.4.7, let L = A + VV. By the Laplacian comparison theorem we have Lp <~ v / K ( d -
1)coth ( v / K ( d -
1)p) - Arp r-
290
Chapter 6
Interpolations of Poincar6 and Log-Sobolev Inequalities
outside of B(0, 1)U cut(o). Since r E [1, 2], there exists so ~> 0 independent of r such that 1( )
Lp ~< - ~
~ - v / K ( a - 1) p~-i =: _c0p~-x,
O f> 80.
Then by Corollary 2.5.4 we obtain
)~(8) ~ C~82(r--1) 4 '
s >~ so,
(6.2.4)
where A(s) denotes the first Dirichlet eigenvalue of - L on B (o, s)C. Then, by e.g. [194, Theorem 1.1], there exists C > 0 independent of r such that (6.0.1) holds. Moreover, (6.2.4)implies 1)] A-l(t)~<(C~Ot) 1/[2(r,
C2 2(r-l) t~> ~ - s 0
Next, for fixed a E ( v / K ( d - 1),A), let W E C ~ ( M ) be such that I [ W - ap[[oo ~< 1 and A W is bounded above. We note that the existence of W follows from Greene-Wu's approximation theorem [95]. Then, according to Proposition 3.4.8, (3.4.5) holds for some c > 0 independent of r. Moreover, it is easy to check that there exists Cl > 0 independent of r such that sup exp[W - V] ~ Cl exp[(A + 1)st], B(o,~)
r > O,
IVWI 2 - I v v I 2 - 2 A ( v - w ) ~< C1 in the distribution sense. Hence Theorem 3.4.7 implies (3.3.3) for/3(s) "= exp[c2(1 + 8-r/[2(r-1)])] for some c2 > 0 independent of r. Thus, it is easy to check with Theorem 3.3.3 that (6.2.1) holds for F(s) [log + s] 2(~-1)/r and some constants C1, (72 > 0 independent of r. Therefore, the proof is completed by Theorem 6.2.1 (1) since it has been shown in the proof of Corollary 6.2.2 that for this F ones has tp ~ c(2 -p)-2(r-1)/r for some c > 0 independent of r. [3 Finally, we present some criteria of (6.0.3) for the birth-death processes and one-dimensional diffusion processes, the proof is completely similar to those of Theorems 5.4.3 and 5.5.1 by replacing log + with (log+) a and using Corollary 6.2.2. T h e o r e m 6.2.4 Let a E [0, 1]. (1) Consider the birth-death process with Dirichlet form given by (1.3.5). There exists C > 0 such that (6.0.3) holds for r = C ( 2 - p)a if and only if n
1
sup j~0 ~ #j n ) l . #jbj j ) n + l
( -log
)o
E #J j)n+l
6.3
Transportation cost inequalities
291
d2 d (2) Consider L "= a(x)-~x2 + b(x)-~x on [0, c~), where a > 0 and a, b are continuous.
'ox a--f-~r~dr.
Assume that # ( d x ) " Let 8 ( f , f ) " -
eC(x)a(x)-ldx is finite, where C(x) "-
#(IVfl 2) with ~ ( s
the completion of Cl(~0, oc)
under the norm 8~/2. Then (6.0.3) holds for r
- C ( 2 - p)~ if and only if
sup
#([x, oc)) (- log #([x, oc))) ~ fo x e -c(y)dy
< c~.
(6.2.5)
x~>0
(3) Let L be as in (2) but w i t h E - R. Then (6.0.3) holds f o r e ( p ) 6 ( 2 - p)~ if and only if (6.2.5) and the following hold: sup
#((-co, x))(- log #((-oc, x))) ~ ~0
-
e -C(y)dy < oc.
x~<0
6.3
Transportation cost inequalities
To understand what the transportation cost is, let us introduce a simple example. Suppose that we are looking at n many cities X l , . . - , Xn, which produce a kind of product. Let p "- ( # 1 , " " , Pn) be the producing distribution of the product, i.e. #i is the amount made in xi, where we assume that the total amount is 1. Now, we intend to transport the product into the demand disd
tribution ~ "- (Ul,"" , Yn), where ~i ) 0 with ~
ui - 1. Thus, # and u are
i=1
two probability measures on E "- { X l , . . . , Xn}. Now, let u be a way of transportation which transports the amount uij from xi to xj, 1 ~ i, j <~ n. If the result of the transportation meets our requirement, n
then one has ~
7rij - vj, 1 ~ j <~ n. Since the initial distribution is #, we also
i=1 n
have ~
~ij - #i, 1 <~ i <~ n. Therefore, a way of transportation is exactly a
j=l
coupling of the measures # and u, and vice versa. Let ~ ( # , u) be the set of all couplings of # and u. To calculate the transportation cost, let p(xi, xj) be the cost rate of transportation from xi to xj. It is reasonable to assume that fl(Xi,Xi) = 0 and p(xi, xj) > 0 for i r j. Then the cost of the transportation u is n
p(x , xj).
i,j=l
pdTr.
j -
xE
292
Chapter 6
Interpolations of Poincar@ and Log-Sobolev Inequalities
Therefore, the optimal (smallest) transportation cost to transport from # to
is
t e a
WlP(#, v) "-
inf
/
7rE~(tt,v),]ExE
pdTr,
which is well-known in probability theory as the L1-Wasserstein distance between # a n d ~. In general, for a metric space (E, p) and two probability measures # and on E, we call WpP(#, ~ ) " =
inf re~(t~,v)
(/E )l/p pPdTr
•
the LP-transportation cost (or LP-Wasserstein distance) between # and v. Our aim is to estimate this quantity by using functional inequalities on Riemannian manifolds. Let M be a connected Riemannian manifold, and let V E C(M) be such that # "- e V(x) dx is a probability measure, where dx is the Riemannian volume element. Let A : T M -~ T M be a continuous mapping such that A(x) is a strictly positive definite, symmetric linear operator on TxM for each x E M. Define s
g) := # ( F ( f , g)) = #((AVf, Vg)),
f,g e C~(M),
where F ( f , g ) " - ( A V f , Vg} for f , g e CI(M). Then (g~, C ~ ( M ) ) i s closable in L2(#). Indeed, we may assume that V and A are COO-smooth since the closability does not change if we replace V and A by smooth V and .4 such that I [ V - ~rllcr < c~ and clA<<..A <<.c2A for some constant Cl, c2 > 0. In the smooth case the Dirichlet form of the diffusion process generated by div(AV) § A V V , which is symmetric in L2(#), is a closed extension to (8, C ~ ( M ) ) . Let (8, ~(d~)) be the closure which is a Dirichlet form on L2(#). Next, let PA be the distance induced by A, i.e.
pA(X, y) -- sup{f(x) -- f(y) " f e C I ( M ) , F(f, f) <<.1} = inf
{j~01V/(A-11s, Is} ds" l. E cl([0, 1]; M), 10 -
x, 11 - y } ,
x, y E M .
Let L := div(AV) + A V V . We assume that (A6.3.1) (M, pA) is a complete metric space. It is easy to check that under (A6.3.1), (E, ~ ( E ) ) is regular and conservative so that the corresponding diffusion process is nonexplosive.
6.3
Transportation cost inequalities
293
To establish a transportation cost inequality for Wp pA, we shall use the following functional inequality for p c [I, 2]" #(f2) _ #(fp)2/p ~< C S ( f , f), f >~ O, f E ~(o~), 2-p where C > 0 is a constant and when p - 2 the right-hand side is set to be (Ip)
(
f ) as p ~ 2. Obviously, (11)is nothing but the Poincard its limit #_f2 log ]]fl12 inequality, while (/2) coincides with the log-Sobolev inequality #(f2 log f2) < 2 C ~ ( f , f) + #(f2)log #(f2),
f e ~(E).
(6.3.1)
Our main tool to deduce the transportation cost inequality from (Ip) is the Otto-Villani's coupling.
6.3.1
Otto-Villani's coupling
In this subsection we assume that V and A are smooth. Since (M, pA) is complete, M with metric gA(X, Y) "- ( A - 1 X , Y) is a complete Riemannian manifold. Let L "- d i v ( A V ) + AVV, and let Pt be the semigroup of the L-diffusion process. Let f e C ~ ( M ) "- { f + C 9 f e C ~ ( M ) , C e R} such that # ( f ) - l a n d e - 1 ) f ) c for s o m e c E (0,1). Let #t "- (Ptf)# which is a probability measure for each t ) 0. Let us fix t > 0. To estimate the Wasserstein distance between #t and #t+~ for s > 0, Otto and Villani constructed a coupling (for A - I) in the following way. Let ~t+~(x) "= V log Pt+~f(x). Then the ordinary differential equation d d--~r - -(A~t+~) o r
r
- I, s ~> 0
(6.3.2)
has a unique solution. We will prove that 7r~(dx, d y ) " - #t(dx)5r
(6.3.3)
provides a coupling of #t and #t+~ which is called Otto- Villani 's coupling, where 5r denotes the Dirac measure at point Cs(x). In this section we prove that r is well-defined and (6.3.3) determines a coupling without any extra condition. To this end, we first prove the following lemma. L e m m a 6.3.1 Under (A6.3.1). For f C C ~ ( M ) with ~-1 >/ f >1 ~, the unique solution to (6.3.2) is nonexplosive with pA(x, Cs(x)) ~ cv/s(s + 1) for some c > O, all x c M and all s >~ O. Moreover, for each s >~ O,r " M ~ M is a diffeomorphism with Cs I satisfying
d d--~r - ( A ~ t + s - u ) o Cu,
r
- I.
(6.3.4)
294
Chapter 6
Proof.
Interpolations of Poincar6 and Log-Sobolev Inequalities
It suffices to prove for noncompact M. Let x E M be fixed, and let n~>l.
T~ "--inf{s >~ O'pA(x, Cs(x)) /> n},
If T~ "-- lim Tn < OC, then there is a sequence {s~} C (0, T~) such that n---+
pA(x, r
(x)
(x)) >~n. But for s < T~ one has
d ds (log(Pt+sf))(C~(x))- -{~t+s o r >/((A~+~) o r
d ~ssr
Pt+sLf - ( Pt+~f ) o r ~+~ o r
-
IILfll . g
Then
fo
~'~((A~t+~) o r
~t+~ o r
<~
IILfll
+ log
-2
Therefore, letting IXIA "-- v / ( A - 1 X , X ) for X e T M , we obtain
n2~pA(X,r
2
(f0sn
d
2
Sn
< Sn
~o
2
((A~s+t) o Cs(x) ~s+t o Cs(x)) ds < IILfll~sn + Sn log S--2 " ~
Letting n --~ c~ we prove that T~ - c~. Moreover, replacing Sn by s we obtain that pA(x, r < cv/s(s + 1) for some C > 0 and all s >~ 0, x e M. Finally, for fixed s > 0, let { r e [0, s]} solve (6.3.4). It is easy to check that r - r Indeed, one has Cs-u - Cu o Cs (resp. Cs-~ = r o r for all u e [0, s], since both of them solve (6.3.4) (resp. (6.3.2)) with initial value Cs (resp. Cs). Hence Cs is a homeomorphism on M. P r o p o s i t i o n 6.3.2 In the situation of Lemma 6.3.1 let Cs solve (6.3.2), then (6.3.3) determines a coupling 7rs for #t and #t+s, i.e. 7rs E ~(#t, #t+s).
Proof.
It suffices to prove that for any h E C~(M) one has
Mh or
= / M hd#t. .
Letting hs "= h o Cs 1, we have hs o Cs - h and hence, d d---~hs - (A~t+s, Vhs) - 0.
(6.3.5)
6.3
Transportation cost inequalities
295
Since h has compact support, by Lemma 6.3.1 and the completeness of (M, PA), there is a compact set B such that hr[B ~ - 0 for all r E [0, s]. Thus, by the symmetry of L,
d /M hrd#t+r - ~rr d /B h r Pt +r f d # - /B [(Pt+rf) "~r d h r + h r (L Pt +r f ) ] d # ~rr - - / B [ d hr - (A~t+r, Vhr)] d#t+r - O. Therefore, (6.3.5) holds.
6.3.2
T r a n s p o r t a t i o n cost inequalities
T h e o r e m 6.3.3
Assume (A6.3.1). Let p e [1, 2]. If (Ip) holds then
W;A(f #, #) ~ P~ C(#(f2 -2/p) p
1)
f >~ 0, # ( f ) - 1,
(6.3.6)
1
where we set (#(f2/p)_ 1 ) / ( 2 - p ) - ~ # ( f log f) for p - 2. Consequently, (Ip) implies 1)
W;A((Ptf)#,#) ~ pe-t/c~/
2-p
V
f~>O, # ( f ) - - l ,
t>~O.
(6.3.7) To apply Otto-Villani's coupling, let us first observe that it suffices to prove (6.3.6) for f E C ~ ( M ) with s-1 ~> f >~ s for some s E (0, 1). Let o E M be a fixed point, and let po(X)"- pA(x,o),x e M. If (Ip) holds for some p e [1,2), then it also holds for p - 1 (see Proposition 6.3.8 below). Hence there is c > 0 such that #(e ~p~ < c~, see Theorem 1.2.5. Thus, for any f ~> 0 with # ( f ) - 1 and #(f2/p) < (x), we have
#(f ~ ) ~ #(f2/P)P/2#(p2oP/(2-P))(2-P)/2 < oo. Next, if (Ip) holds for p - 2 (i.e. (6.3.1) holds), then there is ~ > 0 such that #(e ep2o) < c~ (see Corollary 5.3.2). Consequently, if # ( / l o g f) < c~ then
1
#(:p2o) <~ -~#(f log : ) + p(e Ep2~ <
(:X).
Therefore, we conclude that if (Ip) holds and the right-hand side of (6.3.6) is finite, then #(fpPoo) < c~. Now, for nonnegative f with # ( f ) - 1 and #(fpPoo) < c~, we may choose a sequence of nonnegative functions {f~} C C ~ ( M ) such that n >~ fn >~
296
Chapter 6
n-1 # ( f n ) -
1 and
lz(f2n/P)- 1
fn# -~ f #
n --, ec. Then
Interpolations of Poinca% and Log-Sobolev Inequalities
#(f2/p)_ 1
2-p weakly and
2-p
and # ( I f n -
fl~)
--* 0 as
lim lim #(fnPPool{po>~)) = 0 .
r - - - , (:x:) n - - - ~ (:x)
Hence W ; A (gn#, #) -'~ W ; A (f #, #) as n ~ oe, see e.g. [157]. Therefore, in the proof of Theorem 6.3.3 we may assume that f E C~(M) with ~-1 ~> f ~> for some ~ E (0, 1).
Assume that V and A are smooth and (A6.3.1) holds. For strictly positive f E Cc~(M), we have
L e m m a 6.3.4
d
d~#((Ptf)
2/p)
= -2(2 -
p)8((Ptf) 1/p, (Ptf) 1/p)
1 ~
~#((Ptf) log Ptf) = -48((Ptf) 1/2, (Ptf)l/2). Pro@
We only prove the first formula since the proof of the second is similar. Let B~(o)'- {x "pA(x, o) <~ r} for r > 0 and o E M, and let a on OB~(o) be the ( d - 1)-dimensional measure induced by # and the metric PA. We have
fo c~ dr ~OB~(o) v / F ( P t f , Ptf)da = fM V/F(Ptf ' Ptf)d# <. v/e(Ptf, Ptf) <. V/e(f, f ) < e~, where 8(Ptf, Ptf) <. 8(f, f) follows from the spectral representation of Pt. Since (M, PA) is complete, Br(o) is compact for each r < 0. Then for fixed t > 0/ , there is a sequence of compact normal domains {Bn } such that Bn ---*M and
] J0
IVPtflAda ---*0 as n ~ oc. Therefore, Bn
d L (Ptf)2/pd#- 2_; (ptf)(2_p)/pLptfd# dt ~ P n = - 2(2
-p)
n
F((Ptf)l/v, (Ptf)l/p)d# + P
Bn
(ptf)(2-p)/p(Nptf)da,
where N denotes the outward vector field of B~ with INIA = 1. Since f E C~(M) and is strictly positive, this implies that for some constant C > 0,
d /B r~(Ptf)2/pd# + 2(2 -- p) /B ]V(Ptf) 1/p J ~ d ~ -~, n
<.C f o Bn
V/ F (Pt f , Pt f )d a ~ 0
(6.3.9)
as n ~ co.
6.3
Transportation cost inequalities
297
On the other hand, by the mean-value inequality one has sup Idt d (Pt+,f)2/P ;l l(gt+,f)2/P - (Ptf)2/Pl <~re[t,t+~] =
I
sup - (rt+rf)(2-P)/PPt+rLf ~ C r~[t,t+s]P
for some C > O. Then
8
n
Combining this with (6.3.9) we conclude that and the derivative is given by (6.3.8).
#((Ptf) 2/p) is differentiable
in t E:]
Proof of Theorem 6.3.3.
Since continuous functions can be approximated uniformly by C~176 we may assume that V and A are smooth. Let #t "- (Ptf)# for a nonnegative function f e C~(M) with # ( f ) - 1 and c -1 > / f >~ r for some ~ > 0. By Proposition 6.3.2 and (6.3.2) we have
1 WppA(#t,#t+s)p ~< 71 /L pA(x, r s~
d
dr') p (dx)
< -1 ~ s dr /M (AVPt+,f , VPt+,f) p/2 (r s (Pt+,f)P _ 1 I'M d#t fO~ (AVPt+~f ' VPt+~f)P/2 o Crdr. 7 : -(PT+-,T# Since
/M { sl fo~ (AVPt+,.f , Vpt+,.f)p/2 (Pt+rf) p
o Crdr
}2/p d#t
/M { I ~o0s (AVPt+r/' VPt+rf) Crdr}dpt s (Pt+~f)2 _1_ fo ~dr /M {AVPt+~f ' VPt+~f} d# s (Pt+~f) 1 ~0s E(Pt+,f, Pt+,f)dr <, -8(f, 1 <~-f) < oo, --
g8
0
s
we conclude that when p E [1, 2),
{ l_s~s (AVPt+rf , VPt+rf}p/2(Pt+rf)
e [0,1]}
(6.3.10)
298
Chapter 6
Interpolations of Poincar6 and Log-Sobolev Inequalities
is uniformly integrable with respect to #t. Then for p E [1, 2), (6.3.10) implies that lim WPPA(#t' #t+~)P ~< /M d#t lim -1 fO0~ (AVPt+~f, VPt+~f) p/2 o Crdr
s~O+
sP
s--*O+S
-/M
(Pt +r f )P
( A V P t f , VPtf> p/2 (ptf)p_l d#.
(6.3.11)
Combining this with (Ip) we obtain
!W;A(#t,#t+s ) <<.
(AVPtf , VPtf}P/2d# (Pt f )p-1 <<.p~((Ptf) l/p, (Ptf)l/P) 1/2
lim s~O+ S
<. ; v / c ( :
=
-
- p)
8((Ptf) 1/p , (Ptf) 1/p) V/#((Ptf)2/p) -- 1 -1
v/2 - p dt
Therefore, d+
(.,
dt { -
~<s~o+limslW;A(#t'#t+s) _
.
-
lira
WpPA (# , # t ) - WfJ A (# , #t+s)
s--*0+
p v ~ d-t dv/ <<"- x/'2-----~p
8
(6.3.12)
#((Ptf)2/P)- 1.
Since Ptf -~ # ( f ) i n L2(#) as t ~ cr (indeed, according to Theorem 4.3.1 and its remark, there holds a weak Poincar~ inequality), one has #((Ptf) 2/p) ~ 1 and hence #t ~ # weakly as t ~ oc. Moreover, since (Ip) implies the Poincar~ inequality which in turn to imply the exponential integrability of PA with respect to # (see Corollary 5.3.2), then the weak convergence of #t gives us WppA(#t, #) --* 0 as t -~ c~. Therefore, by taking integral over [0, c~) with respect to t, we prove (6.3.6) from (6.3.12) for p E, [1, 2). To prove (6.3.6) for p - 2, it suffices to show that W~A(f#,#) lira WppA (f#, #) (and hence they are equal). Indeed, for any bounded mea-
p--,2
surable functions g and h satisfying
g(x) <~ h(y) + pA(x, y)2,
x,y E M ,
we have
g(x) <~ h(y) + PA (x, y)P (2N) 2.p ,
x,y E M,
6.3
Transportation cost inequalities
299
where N := 1 + sup{[h[ V [gl}. Then, by the Monge-Kantorovich dual formula of the Wasserstein distances (see e.g. [157]), we obtain lim WpOA(f#, #)P ) #(f g) -- #(h)
p---~ 2
and hence lim WppA(f#, #)P >t W~A(f#, #)2 since g, h are arbitrary. Finally, p---~2
(6.3.7) follows from (6.3.17) below. The following is a direct consequence of Theorem 6.3.3. Corollary 6.3.5 Under (A6.3.1). /f (6.0.3) holds for r some c~ E [0, 1] and some constant C > O, then
WpPA(f#' #) <~P
2/P) 1) (2 -- p)2-~ '
IC(#(f
UI - C(2 - p ) a for
p e [1, 2), f /> 0, #(f) -- 1. (6.3.13)
Finally, we have the following concentration result induced by the transportation cost inequality (6.3.13). P r o p o s i t i o n 6.3.6 inequality
Let c~ e [0, 1]. If (6.3.13) holds, or if the even weaker
W~ A(f#,#) <~ inf p pe[1,2)
1) (2 __ p)2--c~
f ) O, #(f) - I
'
(6.3 14)
holds, then for any measurable set B with #(B) > O, one has #(B~) "- #({x " pA(x,B) >~h}) {1((2-p)(2-a)/2h ~< inf 1+ pC[l,2) C p
) 2 }-p/(2-p) _
C ( 2 -- p)
log # ( B ) - 1
p#(B)(2-p)/p
+
(6.3.15)
where a+ := max{0, a} for a c R. Consequently, 1 (h - v/2C log #(B) -1)2+ ] exp [ - ~-~ <
/ f a = 1,
{ (1+ v/1-a - ~C(a)h-2/(2-P))2} #(B)C(a)h_2/(2_~ ) log #(B) -1
-h2/(2-~)/C(a) +
if h >~C(a) (2-a)/2, 0 <<.a < 1, where C(a) "= ( ( 1 - a)C2a) 1/(2-~) . In particular, limsuph 2/(2-~) log#(B~) ~< -1
C(a)
log(2 - a),
300
Chapter 6
Interpolations of Poinca% and Log-Sobolev Inequalities
where the right-hand side tends to Proof. Then
1 -~-~
as a --, 1.
~ B) for a measurable set B with positive measure. Let #B "-- #(.#(B)
W( A(#g'#hB~) >~h for any h ~> 0. Thus, it follows from (6.3.14) that h <<.W (A (#B, ~) + W( A (#B~, #) <<"P l C(#(B)(p-2)/p _ p)2-a
1)
+ P l C(#(B~)(p-2)/p _ p)2-a
1) .
(6.3.16)
Since there is ~ C [#(B), 11 such that
1 - 2 -p~(p_2)/p log ~-1 ~ 2 - p (B)(p_2)/p log #(B) -1,
p(B) (p-2)/p -
P p (6.3.16) implies (6.3.15). When a - 1, by letting p--~ 2 and using (6.3.15), we arrive at 1 ( h _ v / 2 C l o g # (B)_ 1)+J. #(Bh)c <~exp [-- ~-~ If 0 ~< a < 1 and
h >~ C(a) (2-~)/2, then taking p E [1, 2) such that 2 - p = P
C(a)h -2/(2-p), we obtain from (6.3.15) that #(B~) E { ( 1+
v/1 - a - I
C(a)h-2/(2-p) )2 } -h2/(2-~)/C(~) #(B)C(~)h_2/(2_~) log#(B) -1 + [3
6.3.3
Some results on
P r o p o s i t i o n 6.3.7
(1)
(Ip) is equivalent to
#((ptf)2/p) _ #(f)2/p 2 -p
(Ip) < e -2
/c
_
2 -p
'
(6.3.17)
t >>.O, f >>.O, f c L 2/(pA2)(#). (2)
(Ip) implies (11) for any p >1 1.
Proof.
We only prove for p r 2. In this case the first assertion follows by observing that (Ip) is equivalent to
d 2 d_..~{#((ptf)2/p ) _ #(f)2/p} ~ __~{#((ptf)2/p)
_
#(f)2/p}
for nonnegative f E C~(M) when p >f 1 but r 2. The proof of (2) is also standard by using Taylor's formula. Let (Ip) hold for some p > 1 but r 2.
6.3
Transportation cost inequalities
301
For any nonnegative f E C ~ ( M ) , applying (Ip) to 1 + t f for small t > 0, we obtain
C t 2 8 ( f ' f ) >1
#((1 + t f) 2) - #((1 + tf)P) 2/p 2- p
1 + 2t#(f) + t 2 # ( f 2) - {1 + p t # ( f ) + p ( p - 1)t2#(f2)/2 + O(t3)} 2/p 2-p
= t2[p(f2) _ #(f)2] + O(t3). Letting t --, 0 we obtain (Ip) for p -
1.
V1
We hope that (Ip) is monotone in p in the sense that (Ip) implies (Iq) whenever p >/ q. But in the moment we only have the following slightly weaker statement.
Proposition 6.3.8 (I;)
Let p E [1, co) and consider the functional inequality p[#(f2) _ #(fp)2/p] <~ CoZ(f, f), f ~> 0, f E ~(oz), 2 -;
where C > 0 is a constant. Then (Ip) implies (Iq) for p ~ q. Proof.
Let f c C ~ (M), f >t 0. It suffices to prove that p[(#(f2) _ p(fp)2/p] r
2- p
-
is nondecreasing in p for p >~ 1. Indeed, r (2 - p ) # ( f P ) ( 2 - P ) / P p ( f P l o g
fp
.(IP)
~> 0 if and only if
) ~< p [ # ( f 2 ) _ p(fp)2/p].
(6.3.18)
Since (6.3.1s) is homogeneous in f , we may assume that ~ ( f ~ ) = 1. Then
(6.3.18) follows by noting that (2 - p)r p log r p <. p[r 2 - r p] for all r > O.
[--1
In the next result we present a sufficient condition for (Ip) using the gradient estimate of semigroups.
Proposition 6.3.9 a function ~ ' ( 0 , c o ) ~ IvPt/l~
Let p E [1, 2] and q - 2 p / ( 3 p - 2). Assume that there is (0, ec) with
/o
<~ ~(t)q/2Pt]V fl qA,
th~n (I~) hole~ fo~ C - 2 fo ~ ~(t)dt.
((t)dt < oc such that
t < o, f ~ C ~ ( M ) ,
(6.3.19)
302
Chapter 6
Interpolations of Poinca% and Log-Sobolev Inequalities
Proof. It suffices to prove for p E [1, 2) and strictly positive f E C ~ ( M ) . By (6.3.19) we have
[VptfPl2A -- (IVPtfPlqA)2/q <<.5(t)(PtlV fPlq) 2/q p25(t)[Pt(lv flq f(p-X)q)]2/q <<.p25(t)(PtlV fl2)(PtfP)2(P-x)/P. -
-
Then d
#(f2) _ #(fp)2/p = _
2
_-
P -_
#((PtfP)2/P)] dt
((ptfP)(2_p)/p, LPtfP}L2(.)d t
2(2p2-- P) fo~ dt fM (PtfP) 2(1-p)/p IVPtfPl2Ad#
~< 2(:2 - p ) 8 ( f , f)
/o ((t)dt.
D To prove the gradient estimate (6.3.19), let us recall Bakry-Emery's curvature condition. For f E COO(M), let F2(f, f ) " -
-~ 1 {LF(f,
f) - 2F(f, L f)}
Assume that there exists k E C ( M ) such that F2(/, f ) >1 k F ( f , f ) - k[VI[2A,
f ~ C~(M).
(6.3.20)
According to Theorem 5.6.2, the log-Sobolev inequality (6.3.1) (i.e. (Ip) with 1 p - 2) holds provided inf k ~> ~ . Below we consider the case where k is not necessarily uniformly positive. P r o p o s i t i o n 6.3.10
Let q > 1. Assume that A = I and for some k E C ( M )
( R i c - H e s s v ) ( X , Z ) ~ - k ( x ) l X l 2,
x e M, Z e TxM.
If k is bounded below, then iVPtf(x)l q < ( P t l V f l q ( x ) ) ( E x e - q ' fo k(xs)ds)q/q'
f E C~(M)
1 1 where q~ > 1 is such that -q+ -~ - 1 , and E z denotes the expectation with respect to the distribution of the L-diffusion process starting from x. Consequently, if in addition C(k) "- sup
EZe
fo k(zs)ds
X
2q then (Ip) holds for C - 2C(k) and p - 3 q - 2"
d t < oc,
6.3
Transportation cost inequalities
303
Proof.
Let {Vt c T~tM } be the derivative process of the L-diffusion process {xt}, one has (see e.g. [84], [175])
E/0 k(x,)ds ]
IY~l-< Iv01 ~xp -
a.s.
and
f EC~(M).
( V P t f , Vo) = e ( V f (xt), It),
[:]
Then the proof is completed by using HSlder's inequality. Finally, we consider elliptic diffusions on I~d. Let d d L a , b - E aijOiOj + y~biOi, i.j=l i--1
where a " - ( a i j ) d x d - - h a * for a C 1 matrix-valued function a, and b "- (bi) is a C 1 vector-valued function. 2p P r o p o s i t i o n 6.3.11 Let L - na,b. For p E [1, 2), let q = 3p---~-2" If
G :=
sup
{(v, ova) + II0~(x)ll~s +
X,VG][~d, Vl--'l
4(p - 1)I(v, Ova(x))[ 2} < ~ ,
2- p
(6.3.2~) then Ivptfl q <
(PtlVflq)eqK~t,
t/> 0, f e c~ (R~).
We use a coupling method due to [48]. Consider the following couProof. pling operator of L (see [48]):
L(x, y ) ' - L ( x ) + L(y) d
+ ~
02
(~(x)~(y)* + ~(y)~(x)*)~j Ox~Oyj'
(x,y) E I~d X R d.
i,j=l
Let p(x, y ) " - I x -
Yl. For any f e C2[0, oc) and any x :/: y, according to (2.8)
1 in [48] we have (note that the diffusion matrix in [48] reads ~a rather than a)
L f o p(x, y) - fit(x, y)f" (p(x, y) ) + f'(p(x, y)) (trA(x, y) - ft(x, y) + B(x y)) x-y and z, "- sx 4 - ( 1 - s)y, one has (caution: 2C(x,y) Ix-yl in the definition of A(x, y) in [48] should be replaced by C(x, y)4-C(x, y)* in case that C(x, y) is not symmetric)
where for v "-
304
Chapter 6
A(x,y ) ' <
Interpolations of Poincar~ and Log-Sobolev Inequalities 1
Ix - ~12 (~ - y, ( ~ ( ~ ) - ~ ( y ) ) ( ~ ( ~ ) 'IL,~
- ~ ( y ) ) * ( ~ - y))
|
I~-yl 2 /o 1 I(~.o.~)(zs)12ds.
trA(x, y) +/}(x, y ) " - tr((a(x) - a ( y ) ) ( a ( x ) - a(y))*) + (x - y, b(x) - b(y)} <. I~ - yl 2 {j~oI (llOv,,ll~s(z~)
Letting f ( r ) - r q' for q' = obtain Lp q' (x, y) <~ qtpq'
(x, y)
qtKqpq'
2p 2-p
{(q' - 2)1(~. o~)12
Thus, by a standard argument
x,y
v)(Zs))ds
).
which is the conjugate number of q, we
~1
(x, y),
+ (Orb,
+
IIo~ll~s +
(O.b,,> } (z~)ds
EN d.
we arrive at
EX'Yp q' (xt, Yt) < pq' (x, y)e q'Kqt ,
t~>O,
where (xt, yt) is the coupling process generated by L. Therefore,
IPt f (x) - Pt f (y) [ <
ix-vl
<
x
'
(E~,, IS(x~) p~(~: ~ )S(y~) - I~)l/q (E 'YpqixzYi(Xt, Yt))
1/q'
eK~t(E~,yIf(xt)pq(xt(y_~ f(Yt)lq ) 1/q
Letting y --. x we have yt ~ xt a.s. and hence the proof is completed by using Fatou's lemma. E] 6.4
Notes
This chapter is reformulated from [206] and [213]. The inequality (6.0.3) was first studied by [121] for r C ( 2 - p)~ with a e [0, 1] and some constant C > 0. In that paper they proved that this inequality holds for a - 2 ( r - 1)/r
d
when E
[xi[r] i--1 with Z the normalization. This result is generalized by Corollary 6.2.3. The inequality (6.1.2) is well-known as Deuschel-Stroock's inequality which follows from (6.1.2) by first dividing by 2 - p then letting p ~ 2. In Corollary 6.2.2, the equivalence of (2) and (4) is observed by [184] and that of (1)-(3) is due to [213]. -I~ d
and 8(f, f ) " -
#(IV f[2), where # ( d x ) " - Z exp [ - ~
6.4
Notes
305
The transportation cost inequality was initiated by [174]. Talagrand proved for the standard Gaussian measure # on R d that
W2(f #, #)2 ~ 2#(f log f),
f~>0, #(f)-l.
This result is then extended to the Riemannian manifold setting by [150] using coupling, and [21] using Hamilton-Jacobi equations. More precisely, they proved that if there is a constant C > 0 such that
#(f2 log f2)
<~ 2c#(ivfl2),
f e C~(M), #(f2)_ 1,
(6.4.1)
then
W2(fp, p)2 <~2C#(f log f),
f ~> 0, # ( f ) - 1,
(6.4.2)
where the W2 is defined with respect to the Riemannian distance. This result is now covered by Theorem 6.3.3. [92] also studied the transportation cost inequality with Wp for p > 2 using stronger functional inequalities. Moreover, letting (E, p) be a metric space, [75] proved that the following inequality W~(#,
f#) <. v/2C#(f l o g / ) ,
f / > 0, # ( f ) - 1
(6.4.3)
holds for some C > 0 if and only if (# x #)(e ~p2) < oc for some A > 0. Their proof is based on a result due to [22] which says that (6.4.3) is equivalent to #(e )'(F-"(F))) ~< exp[A2CllF[[2Lip/2],
A e R,
where II. IIL~pis the Lipschitz constant of a function with respect to p. It is now interesting to find out examples on manifolds with # " - e Vdx for each of the following: (i) (6.4.3) holds for some C > 0 but (6.4.2) does not hold for any C > 0. (ii) (6.4.2) holds for some C > 0 but (6.4.1) does not hold for any C > 0. According to [200] and [75], there exist examples such that (6.4.3) holds for some C > 0 but (6.4.1) does not hold, so that one of (i) and (ii) must be satisfied. But it is not clear which one is satisfied. Moreover, it was proved by [92] that the Bernoulli measure, which is however discrete, satisfies (i). Otto-Villani's coupling was first constructed in [150] for A = I and V E C~(M) such that Ric-Hessy is bounded below. Here we extend their construction to a more general setting without this curvature condition. Finally, Proposition 6.3.8 is due to [121]. But the monotony of (Ip) in p is still open.
Chapter 7 Some Infinite D i m e n s i o n a l Models In this chapter we establish functional inequalities for Markov processes on some infinite-dimensional spaces, including the (weighted) Poisson space, abstract Hilbert space and the path space over a Riemannian manifold.
7.1
The (weighted) Poisson spaces
Let E be a Polish space with ~ the Borel a-field, and let # be a a-finite measure without atom. Consider the configuration space n -
i=1
equipped with the a-field induced by {7 ~ -y(A) 9 A E ~ } , where 5x is the 0 Dirac measure at x and ~ 5z~ "- 0 is the zero measure by convention. By the i--1
consistence theorem (cf. [151, Theorem 4.2]), there exists a unique probability measure 7r~ on F such that n
7r~('~(A~) - ki 1 ~ i <<.n ) " - H - e-t~(Ai)pt(Ai)ki ) ki! i=1 '
for any n ~> 1 and any disjoint sets {A~}~ 1 c ~ with #(Ai) < c~. This measure is called the Poisson measure with intensity #. Since # does not have atom, 7r~ has full measure on F0 "= {7 e F 9 V({x}) ~< 1 for all x e E}. In this section we first study the weak Poincar@ inequality for the second quantization Dirichlet form, then describe functional inequalities for a class of finite particle systems. 7.1.1
W e a k Poincar@ i n e q u a l i t i e s for s e c o n d q u a n t i z a t i o n Dirichlet forms
Let (80, ~(80)) be a symmetric Dirichlet form on L2(#) with associated subMarkov semigroup Pt. To determine the second quantization of Pt, let us
7.1
The (weighted) Poisson spaces
307
recall the It6 map" In" L I ( # n ) ~ L 2 ( # n) ~ C n, the n-th chaos of the Poisson space. Given f e nl(# n) ~L2(#n), let
In(f)(7) "-- /E f ( X l , ' ' " ,Xn)(7 - # ) ( d x l ) ' " ( 7 - #)(dxn), where E.n "= {(Xl,"" ,Xn) e E n ' x i
.1 y. ~ .f(xo-1, . n (T One has
~ xj for i ~: j}. Let f ( x l , . . . ,Xn)"=
, XO-n) with a running over all rearrangement of (1..., , n).
7%~(In(f)In(g))
-
-
5n,m#n(f g),
f, g E L I ( # n) N L2(pn)'
where 5m,m := 1 and 5m,n : = 0 for n # m. Now, the second quantization semigroup pO of Pt is a symmetric Markov C0-contraction semigroup on L2(Tru) determined by (see [173], [144] for more details) -- 1 ~ pF'OT -t -n(f)"
P/'~
In(p2nf),
f E L2(#n) N L I ( # n ) ,
n ~ 1.
Let (g~ , ~(E0r)) be the associated Dirichlet form on L2(Tru). It is well-known that (see e.g. Simon [165]) gap(8or) - Ao(E)"= inf{g~o(f, f ) ' f
e ~(~o), # ( f 2 ) = 1},
which is exactly the bottom of the spectrum o f - L , where L is the generator of Ft. To study the weak Poinca% inequality, let us introduce A(r) : - i n f { 8 o ( f , f) " f e ~(8o), p(f2) _ 1, ]If IlL < r},
r>0.
T h e o r e m 7.1.1 A(r) > 0 for any r > 0 if and only if there exists a : (0, oc) ~ (0, oc) such that (F
-
(F)
<
F (F.
F) +
2.
F E @ ( S f ) , r > 0, (7.1.1)
where 5(F) := sup F - inf F. More precisely: (1) If (7.1.1) holds then
A(c) ~ sup 1 - - 4 4 r ( 3 + C 2 ) ( 2 + 8 - + - 8 -1 ) > 0, s,r>o
C > 0.
(1 + 8-1)oL(r)
(2) If A(r) > 0 for all r > 0 then (7.1.1) holds for a(r) "- )~(r-1) -1 Consequently, one has gap(g~ r) - A ( c r A0(g~
Chapter 7
308
Some Infinite Dimensional Models
To prove this theorem, let us first present the following two lemmas. L e m m a 7.1.2 For any f e LI(#) ~ L~(#), let F(~/) "- ~/(f). Then Try(F) #(f), r ~ ( F 2) _ #(f)2 + #(f2) and 7rt~(F4) - #(fd) + 3#(f2)2.
Proof. The desired results follow immediately from the Laplace transformation of r~" pe'~(f)~rt~(d"/) - exp[#(e f - 1)],
f e Li( )
see the proof of [158, Lemma 7.2] for details.
[~
L e m m a 7.1.3 If F e ~(8o F) then for ~ - a . s . ~/, D.F(~/) := F(~ + 5 . ) F('~) E ~e(E0), the extended domain of ~(8o), and
s
(F, F) - ]r E(n.F(~/), n.F(~/))~(d~/).
(7.1.2)
Proof. Let ( / , ~ ( L ) ) and (iFo,~(i~)) be the generators of Pt and Ptr'~ respectively. Let 9- s p a n { F " F -
In(f | for some n/> 1, f E ~(5)}. Then ~ is dense in i2(Tr~) and stable by Pt, see e.g. [144, p. 58]. Thus, ~ is a 1/2 core of ~(L0F) under the graph norm and therefore of ~(80 F) under (5~0F)1 . Moreover, it is easy to see that (7.1.2) holds for F E %0 and therefore, holds for all F E ~(E0F) by an approximation argument, see [219] for details. K] Proof of Theorem 7.1.1. Assume that (7.1.1) holds. For any f E 5 1 ( # ) ~ ice(#) N ~ ( 8 0 ) with #(f2) = 1 and I[f][er ~ c, let F ( ~ ) : = ~(f) and FR "= (F A R) V ( - R ) for 7 e F, R > 0. It follows from (7.1.1) and Lemma 7.1.3 that 7r~(F~) - 7rtt(FR) 2 < a(r)Eor(II(f),Ii(f)) + 4rR 2 = a(r)oZ(f, f) + 4rR 2, Combining this with Lemma 7.1.2 we arrive at 1 -- ~r~(F 2) - Try(F) 2 < 7r~((F - r , ( F n ) ) 2)
r, R > O.
(1 + s)Trt~((F- FR) 2) + (1 + s-1)~,((FR -- 7~t~(FR))2) (1 + s)Trp(F21{IFI>R}) + (1 + s-1)a(r)80(f , f) + 4(1 + s-1)rR 2 < 4(1 + s-1)rR 2 + (3 + c2)(1 + s)R -2 + (1 + s - 1 ) a ( r ) 8 o ( f , f ) ,
r , s , R > O.
Therefore, 1 -- [dr(1 + 8-1)R 2 + (3 + c2)(1 + s)R -2] 8o(f, f) ~> sup
(1 -~- 8-1)o/(r)
r,s,R>O
-
-
sup
1 -- 4 v / r ( 2 ~- 8 + 8 - 1 ) ( 3 + c 2)
(1 +
7.1
309
The (weighted) Poisson spaces
On the other hand, assume that ,k(r) > 0 for all r > 0. For any F e ~(oz0r), by [216, Remark 1.4] we have #((D.F)2)dTr~.
-
Thus, by the definition of ,~(r) and Lemma 7.1.3 we obtain (F
-
(F) 2 <
<
1
1
)~(R) 7r~(o%(D.F,D.F)) + ~rc~({{D.F{[~) 1 1 E0p (F, F ) + ~ 5 ( F ) 2, /-t A(R)
R > 0.
This completes the proof. 7.1.2
[-7
A class of jump
processes on configuration
spaces
Recall that the jump process (or the continuous-time Markov chain) on Z+ describes the behavior of the number of particles for a (finite) particle system. If we also consider the locations of particles, then the state space of the process reduces to the (finite) configuration space F'-
5x~ " ec > n ) 1. xi E E. l <. i <~ n
.
i=1
where (E, ~ , #) is a probability space such that {x} e o~ with #({x}) = 0 for all x E E. We shall study a class of reversible Markov jump processes on F. We first construct the reference probability measure which is a weighted Poisson measure. Since # is finite, the Poisson measure 7r~ has full measure on the present finite configuration space. Moreover, we have oo
7 r , ( B ) - el 1 B ( 0 ) +
~
#n(Bn)en! '
B E o~r,
(7.1.3)
n=l
where #n is the product measure of # on E n and Bn "- { ( X l , ' "
,Xn) C E ~ :
n i=1
For a strictly positive measurable function 0 on (F. o~r) with 7r.(o) - 1. let 7r,,o(dT) "- ~)(v)Tr,(dv). We assume that Q(7) - O ~l for a sequence of strictly positive numbers {~On}n) 0. For any F C Ll(Trt~,o), let n
F (0) "- F(O),
F (n)(xl,
,Xn) "-- F ( ~ ( ~ x i ) , i=1
xi C E , n >~ 1, 1 <. i <~ n.
310
Chapter 7
Some Infinite Dimensional Models
By (7.1.3) we have F (n) E ~,~n for n ~> 1 and (:X)
(3o
7rtt,o(F ) __ ~ ~n . n(F(n) ) m n=O n=O
(n) ~n(F(n)
),
where m~(n) "- Qn/(en!) gives rise to a probability measure on Z+. We shall construct a symmetric Dirichlet form on L2(Tr~,e) in terms of a Q-matrix symmetric with respect to m~, so that the number of particles for the corresponding particle system is exactly the given Q-process. Let Q " - (qi,j)i,j>~o be a totally stable, conservative Q-matrix on Z+, i.e. qi,j >/ 0 for i r j and --qi,i -- qi "-- Y ~ qi,j < (x) for all i. Let q ( 7 ) " = q(7, F)
j#~ and c~
k
k-1
i--1
k=l
I')'l!
ql'Y]'i'Yl-k ~ E Fk ' ' ) ' - - ~ E A ,
"y E F, A E ~ r ,
where Fk := {'y E F " I-y] "= -y(E) - k}, k >~ 0. To make this q-pair symmetric, we assume that Q is symmetric with respect to me, i.e. (A7.1.1)
qi,jme(i) - qj,im~(j),
i , j >~ O.
7.1.4 (q, q(., d~)) is a totally stable, conservative q-pair on (A7.1.1) holds then q(., d-y) is symmetric with respect to 7r,,~, i.e. 7r,,e(d-y ) q(-y, dr/) = 7r,,e(d~)q(~/, d-y). Proposition
(r,
Proof. (a) To prove that q(-y, d~) is a well-defined transition probability on ~ r , let us recall a classical fact. Let (~,~,~i,#i)(i = 1,2) be two measure spaces. For any B E ~1 x ~2, one has B~ "= {y E E 2 " ( z , y) E B} E ~2 and n
#2(Bx) is ~'l-measurable in x. Now, for any A E ~ r and -y = ~
5y~,one has
i=1
An-kk E ~,~n-kk
and
hence, k
,Xk) E Ek " ~/ + i-1
= {x E E k ' ( x , y) E fin+k} --(An+k)y E ~,~k, where y - ( y l , " " ,Yn). Moreover, by (7.1.3) we have #k(c~) -- ek!Tr,({~ E Fk ' ' y § E A}). L e t B := {(-y,~) E F x F " ~ E F k , ' y + ~ E A}. Then it
7.1
The (weighted) Poisson spaces
311
is easy to see that B E o~r x o~r and hence pk(c,y) -- (ek!)1r~(B~/) is o~rmeasurable in 7. This means that the first term in the definition of q(v, A) is well-defined and is a transition measure. So, it remains to show that ~A (~) := #{U E Fk : V - U E A} is o~r-measurablefor any k ~> 1. Let B := {(V,U) : F x F " rI C Fk, 7 - r/C A}. Then B E o~v x o~r and hence ~A (')/) -- 5//L(B~,) is o~r-measurable. (b) Since Q is totally stable, we have oc
I~1
q(~) . - q(~, r ) - ~
q ~ , ~ +~ + ~
k=l
q ~ , ~1-~ - q ~1 < ~ ,
~ e r.
k=l
Then the q-pair is totally stable and conservative. It remains to prove the symmetry of the measure J ( d ' h , dff2) := 7rtt(d')'l)q("/1, d"/2). For any m > n and measurable sets An C Fn, B m c Fro, we have
/A
J(dn x Bin) -- qn,m
#m-n
(:( X l , " "
rn--n }) ,Xm-n) "~ Jr- ~ ~xi e Bm
n
7rp,0(dv)
i=1
#m-n({x E o~m-n'(y,x)
= qn,mmo(n )
E Bm})l.tn(dy)
n
= q n , m m o ( n ) # m ( ( A n • E m - n ) A Bin).
(7.1.4)
On the other hand, since 7r, (and hence 7r,,0) has full measure on F0, we may assume that V considered below does not have multiple, i.e. V({x}) ~< 1 for all x E E. Thus, J(Bm x An)
qm,nn!(m- n)!mo(m)
?Tt!
1 ~ (Xil, . . o , x i n ) # m ( d x l , . . . , dxm).
/[~ml~il<~~in~rn
Since .4~,/)m and #m are symmetric in coordinates, one has
J(Bm • An) = q m , n m o ( m ) f ~ JIb
m
l ~ n ( X l , ' ' " ,Xn)#m(dxl, "'" ,dxm)
= q m , n m o ( m ) # m ( ( A n • E m - n ) N Bin).
Combining this with (7.1.4) and (A7.1.1), we obtain J ( A n x B m ) - J ( B m x A n ) . Therefore, for any measurable sets A and B, letting An " - A A Fn and Bm := B N Fro, we otain oo (Do J(A x B) - E J(An x Bin) - ~ J(Bm x An) - J(B x A). n,m=0 n,m=O
312
Chapter 7
Some Infinite Dimensional Models
Next, we define the form f(F,a) := 1 / r • (F(',/) - F0?))(G('y ) - GO?))Tr..o(d').)q(~/. d~?).
~(s
"= {F e L2(r,,o) 9 s
F) < c~}.
7.1.5 Assume (A7.1.1). Then (8fl, ~ ( S f )) is a well-defined symmetric Dirichlet form on L2(Tr~,o) with generator
Proposition
L~"F(-).) " - / F (F07) - F('y))q('),. dr/).
where ~(L~) is determined by the Dirichlet form. Moreover, we have
Qnqn,m / r r r . ( d T ) f E m _ ( D z F ( ~ / ) ) ( D x G ( @ ) # m - n ( d x ) , n=0
m=n+1 (7.1.5) m-n
where F, G e ~ ( S f ) and DxF(~/) "- F(~/ + ~
5 x i ) - F(~/), x "- (Xl,...
i=1
Xm-n) E E m-n. Proof.
By the symmetry of q(., dT) we have
Ef (F, a) :=
1
(F(@ - F(rl))(G(@ - G(rl))rr.,o(d~/)q(~/, dr/) n,m=O
(2()
X Fm
OO
(F(~) - F(rl))(G(~) - G(rl))rrz,o(d~/)q(~/, d~)
= m=n+l -- ~ n=0
Xl'm
~ Qnqn,m/Fn 7 r # ( d ~ ' ) m--n+1
m-n
(DzF(~/) ) (D~G(~/) )#m-n (dx).
Thus, (7.1.5) holds. We note that D" L2(Tr~,o) ~ L2(# m-n • 7rt~,o) is welldefined since if F = F' (Tr~,o-a.e.) then O F - OF' (#m-n • 7r~,o-a.e.). Next, let %00 "- { F e L2(Trg,o) 9 there exists no >1 1 such that F(7) - 0 for ]71 i> no }. n
It is easy to see that for any F C L2(Tr~,0), ~
FIF~ C %00 -o F in L2(r~,Q)
i=0
as n ~ c~. Then ~0 is dense in L2(Tr.). Moreover, for any F e ~0, by the
7.1
313
The (weighted) Poisson spaces
symmetry of the q-pair one has (:x)
(:x:)
n=O
re=n+ 1
4(p,p) -
qn,m fFr Trtt,o(d'/) / E m _ n ( D x F ( 9 / ) ) 2 # m - n ( d x )
E
no
(2~
{qn,m /I',~ F2drr"'~
-~-m~
}
n=O r e = n + 1 no
<~ 2 sup O~n~no
no
qnTru,o(F2)+2~-~. ~ n=O
#m(F(m)2)< oo.
re=n+1
Then ~0 C ~(ozjr) and hence the form is densely defined. Obviously, the definition of ozf implies the sub-Markovian property, i.e. for any F e ~(ozf) and any function h on N with [h(s) - h(t)l ~< I s - t l, s, t e N, one has h o F e ~(ozsr) and ozjr (h o F, h o F) ~< 8jr (F, F). So, to prove the form is a Dirichlet form, it remains to verify the closeness. Let {Fk }k/>1 C ~(ozjr) be a Cauchy sequence with respect to the Sobolev norm II" [[L2(~,,o)+ ozf(',') 1/2. Then there exists F e L2(Tr,,o) such that Fk ~ F in L2(Tr,,0) as k ~ co. Without loss of generality, we assume that the convergence is also 7r,,o-a.e. Thus, -p(rn)k ~ F(m) (#m_a .e.) for all m ~> 1 and p(0) k "-- Fk(O) ~ F(0) "-- F(O). Therefore, by the Fatou lemma we obtain (X)
(:x:)
n=O
re=n+ 1
(X)
CX:)
4(F,F)-~
~
-- E n=O
qn,mfF dTrt~,OfEm_n(DxF)2#m-n(dx)
E
re=n+ 1
qn,mmo(n)qn'mif'Era(F(m)(X, y)
_ F(n)(y))2#m-n(dx)#n(dy) CX)
O0
-- E n=O
E
qn'mmo(n)qn'ms
re=n+1
lim m k~cc
(F(km)(x, y)
_ p(n)(y))2#m_n(dx)#n(dy ) ~k lim 8f (Fk,Fk), k--~oc
which is finite since {Fk} is a d~ lim 8jr ( F k---*oc
- Fk, F -
Thus (Sjc, ~ ( S f ) ) is closed.
sequence. Similarly, we have
Fk) <~ lim
lim ozjc ( F j k----~oe j ___,cc
- Fk, F j - F k ) = O.
314
Chapter 7
Some Infinite Dimensional Models
Finally, for any F, a e ~(Sjr), by the symmetry of J (d-y, dr/):= 7r~,~(d-),)q(-y, dr/) one has
fl ~
G("/)(F(71)
F(~,))J(d-y, dr/) - - f/~
-
xF
G(q)(F07 ) - F('~))J(d-),, dr/). xF
Then
-Tr~,e(GL~F) =
~
• (G(~/) - GG/))(F('),) - F07))J(d~, dr/) - 8jp (F, G). [3
Therefore, L~ is the generator of 8 f . 7.1.3
F u n c t i o n a l i n e q u a l i t i e s for d~sc
We first consider the spectral gap of (83r, ~ ( S f ) ) " gap(d~sr) "- inf {d~ (F, F)" F E ~ ( 8 ~ ) , 7r,,o(F 2) = 1, 7r,,o(F ) = 0}. Since EjF is induced by the Q-matrix, it is natural for us to relate its spectral gap to that of OZQ9 gap(SQ) 9 inf {SQ(r,r) "r e ~ ( S Q ) , m ~ ( r ) - O , m ~ ( r
2) - 1};
where 8Q is the Dirichlet form of the Q-matrix (under condition (A7.1.1))"
n=0 m=n+ 1
1"-
{rn} , 8
{8n} e ~(EO) "-- {r e L2(mo) " So(r, r) < oc}.
Next, we consider the following weak Poincar~ inequality:
~.~(F 2) < ~j(rDS~ (F. F) + rllFII 2
0. (7.1.6) where a j 9 (0, oc) --~ (0, oc) is a positive function. We shall also connect this inequMity with the corresponding one of 8Q" ~,
m0(r 2) < aQ(r)SQ(r, r ) + r s u p r n 2, r > 0, n~>0
~ > 0. F e ~ ( S f ) . . . . ~ ( F ) =
I"- {rn}n>/O e ~(O~Q), mQ(~')- O.
(7.1.7)
T h e o r e m 7.1.6 Assume (AT.l.1). (1) gap(d~Q) ) g a p ( S f ) ) O~ Consequently, gap(SjF) > 0 if and e only/fgap(ozQ) > O. In particular, for the birth-death case where In " - - q n , n + l >
II
"l::
0
tO
tO
N.,
r~
\V
A
o
~
9
0
II
tO
II
o
o
i...a
r~
0o
o
+
9
to
N
I
+
~
N
I
~
I
U~,/~
~
:~
0
o
in,.
=
~
tO
9
~% ~
---
II
-"
/A
+
9
~_~
..,,
II
r~
--
I,
...
bl
c--ti..-, 9 0
~
~
II
('D
8
"~
._. ~--,
II
~
~~'~'~ "
~
T
~
:A
9
~~
~~
~
to
"~
~
9
~'
~
~
~ ~
9
~
~.
~
9
~
~:)
L'B
~I ~
~
~
~
~~
9
~ ~ ~ .
"q
~ ~
~-~
~
c~
O0
A
8
,.~
~
v
c~
V
~V
m,....i
0
:~ C' D i~,o
('D
."q
316
Chapter 7
Some Infinite Dimensional Models
Thus, for any F E ~(ozf) with 7r~,~(F) - 0 we have
8f (F>1~o(8r
8 f (F.F) =
F(O).F - F(0))/> AO(SQ)Tr..o((F- F(0)) 2)
This implies that gap(d~f) ~> A0(d~Q). Since for a n y r with r0 - 0 and me(r 2) 1 one has m~(~ 2) - m~(~) 2/> m~(~ 2) - m~(~2)(1 - m~(0)) - eo, e 00 A0(SQ) /> e g a p ( S Q ) and hence the desired lower bound of gap(d~f) follows. Therefore, the proof of (1) is finished by Theorem 1.3.11. (b) By taking F "= E rnlFn one concludes that (7'.1.6)implies (7.1.7) for n=0 aQ - az. On the other hand, for any F E ~(o~f) with F(0) - 0, it follows from (7.1.10) and (7.1.7)that "
OLQ(r)~(F, F) ~ 7r#,o(F 2) -
oo
(E
mo(Tt)i"n(F(n)2))2 -~llFllL
n---1
~> 7r.,o(F2)mo(0) -
rl[F[l~.
Therefore, for any F e ~(E3F) with 7r.,~(F)- 0,
e e 7r~.e(F 2) ~< 7r..e((F- F(0)) 2) ~< - - a Q ( r ) E f ( F . F ) + rllF- F(0)II 2 ~0 ~0 e 4er <<.-~Q(r)SfOo (F, F) + -~o IIFII2~" e
This implies (7.1.6) for ~j - - - a Q ( o o r / ( 4 e ) ) . O0 (c) For any nonnegative f e L2(#), let F(~/) "- ")'(f)lrl. 7r#,0(P~-~j -- ea01 aO1 # ( f ) and 7r..o(F 2) - __#(f2) and e
We
have
,~f (F~ F) - --qo,l~( Qo f2 )--[- -Q1 - ~ ql,m~( f2 ) < (aoo V aol)ql#(f2). e e m--2 Thus, if the super Poinca% inequality holds then there exists constant c > 0 such that #(f2) <~ c#(f)2 for all nonnegative f, which is impossible since # does not has atom. K] Finally, we present an example to show that in general gap(Sj r) is strictly less than gap(EQ).
7.2
317
Analysis on path spaces over Riemannian manifolds
Example 7.1.1
Let ql,k
1 -- ~k
> O, q k , 1 - - -~
for k # 1, and qi,j
--
0 for i, j #
1. By (A7.7.1) one has too(1 ) - (1 + 2ql) -1 and mo(k ) - 2mo(1)~k(k # 1), where ql "- ~
ql,k < ce. Then g a p ( E Q ) - 1 (see [65, Example 4.7]). On
k=/=l
the other hand, for f e L2(#) with # ( f ) - 0 and #(f2) _ 1, let F(7) "7 ( f ) l r l (~/). We have ~ , e ( F ) - 0 and 7r~,e(F 2) - m~(1). Moreover, by the symmetry of the q-pair,
8 f (F, F) - Z
qk,lm~(k)
1 (1 - - -~
- too(l))
k#l
Therefore, gap(Ear ) <~ 1 - m e ( l ) 2me(1)
7.2
<
1
-- gap(EQ)
2
1
if m ~ ( 1 ) > 2 -"
Analysis on path spaces over Riemannian
man-
ifolds It was in 1994 when [85] realized the Poincar6 inequality on the finite timeinterval Brownian path space over a compact Riemannian manifold. Fang's observation, in particular his version of Clark-Ocone-Haussmann formula, stimulated a series of sequel papers concerning Poincar~ and log-Sobolev inequalities on path and loop spaces. It is now very clear that the log-Sobolev (and hence the Poincar~) inequality holds on the finite time-interval Brownian path space provided the based Riemannian manifold has bounded Ricci curvature, see e.g. [5], [103], [104], [30], [190] and references therein. The first purpose of this section is to prove the weak Poincar~ inequality on path spaces either with infinite time-interval or with finite time-interval but over unbounded Ricci curvature manifolds. Next, as a continuation of w where the transportation cost inequality was studied on Riemannian manifolds, we intend to establish such inequality on Riemannian path spaces.
7.2.1
Weak
Poincar6
inequality
on finite-time
interval
path
spaces Let (M,g) be a (metric and stochastic) complete Riemannian manifold of dimension d. Let o E M and T > 0 be fixed. Consider the path space MoT "- {x. e C([0, T]; M ) " x0 - o}
318
Chapter 7
Some Infinite Dimensional Models
equipped with the (product) a-field induced by the class of cylindrical functions (T)
=
{F
. F(x
n ~> 1, f E c l ( I ~ n ) , o
) -
f
, . . . ,
),
< 81 < "'" < 8n <
T}.
Let ~T denote the distribution of the Brownian motion starting at o up to time T. Then #T is a probability measure on M T Let {Ut}t>~o be the horizontal Brownian motion on O(M), the bundle of orthonormal frames with Uo -- uo E Oo(M). More precisely, let {Ut}t>~o solve the stochastic differential equation d
d Ut - ~
Hi o dB~,
Uo - u0,
i--1
where {Hi}idl are orthonormal horizontal vector fields on O(M), and Bt ")i=1 is the Brownian motion on IRd. It is well-known that xt - 7r(U~) provides the Brownian motion starting at o, where ~ 9 O(M) ~ M is the canonical projection. For h E H "- {h E C([O,T];R)
:=
" IIhlI
h'(s)2ds < e~}, let Dh
denote the derivative along direction h. F o r F E o~'C~(T) with F(3') f (X81 , ' ' " , XSn ), one has n
DhF(x') - ~-~{Vi f , U~,h~,}(x.), i--1
where V i is the gradient with respect to the i-th component. Then the gradient DF(x.) e g is defined by (DF(x.),h}H := DhF(x) for h e g . This, together with the measure #T, gives rise to a natural quadric form defined on ~ C ~ (T)
8(F, G) " #T((DF, DG)H) - E(DF, DG)H. If
9 ~;IRicx~12ds < ~ , where
IRicx~l denotes
the operator norm of the Ricci
curvature Ricx~ 9 T ~ M ~ Tz~M, then Driver's integration by parts formula introduced in [79] holds. This implies that (D, o~'C~ (T)) is the quadric form of a symmetric operator and hence, the form (8, o~'CI(T)) is closable in L2(# T) and its closure (s ~(d~)) is a Dirichleg form on L2(#T). Let (PtT)t>~o be the associated (Ornsgein-Uhlenbech) Markov semigroup. We consider the following weak poincar~ inequality" such that E F 2 < a ( r ) ~ ( F , F ) + rllFII 2oc,
r>0,
EF=0,
Fe~C~(T),
(7.2.1)
7.2
Analysis on path spaces over Riemannian manifolds
319
where a : (0, c~) --+ (0, c~) is a positive function. Let K be an increasing function such that
xEM,
IRicx] ~< K(p(x)),
where p(x) : - dist(x, o) is the Riemannian distance. nonnegative and increasing such that Ric(X, X ) > / - K 1 (p(x)),
x E M , X e TxM,
(7.2.2) Moreover, let K1 be
IXl-
1.
(7.2.3)
Finally, for R > 0, let 7R := inf{t >~ 0: p(xt) >~ R}. T h e o r e m 7.2.1
/f lim 1 f~ ds R---,oc v/P(TR ~ T) J R eTKI(s)/2K(s ) = c~,
(7.2.4)
Then (7.2.1) holds for a(r) "- inf {8 + eTKI(R)K(R)2} < oe,
REAr
r>0,
where
A~.-
R>0.
inf
P(TRI<~T)
R1E(0,R)
8T + T 3
3+
/R~1K(s)le -TKI(s)/2 , ds )2
~ r
(a) The main tool of the proof is the following Clark-Ocone-Haussmann Proof. type formula observed by Fang in [85] for compact Riemannian manifolds: F - EF +
F e ~,~C~ (T),
(7.2.5)
where
H E "- E DsF + ~ d Here D~F := ---~(Dg)*' 0d ~
~5sl
qStRicut( D t F ) d t l ~ s .
is the natural (#T-complete) filtration of B,,
Ricv, (v) E IRd with v c IRd is defined by (Ricv,(v), v'} "- Ric(U~v, U,u') for all v' E I~d, and 45t solves the equation dqSt 1 d-T + 2 ~tRicvt --O,
~0 = I.
320
Chapter 7
Some Infinite Dimensional Models
To apply this formula for the present setting, we make a local version with unbounded IRic I. For R1 > O, let
=
r
"
1
/R ~lVr
v/P(TR 1 ~ T)
1
d8
eTKI(s)/2K(s),
r ) O.
For R > R i, let r
'
0.
Then hR(r) -- 1 if r ~ R1 and hR(r) - - 0 for r/> R. Next, let fR e C ~ ( M ) be nonnegative such that fR = 1 on B(R) "{p ~< R}. Let MR := {fn > 0},gR "-- fR2g on MR. Then (MR, gR) is a stochastic complete Riemannian manifold with Ricci curvature Ric R bounded, see the proof of Proposition 2.1 and Proposition 2.3 in [176]. Let (MR) T "-{~ e C([0, T]; M n ) ' x o -- o}. Then we have the following Fang's formula on (MR)To because (7.2.5) holds whenever the curvature is bounded (see [30])"
F dBs>, F - ERF + ~o T (H n,~,
F e 9vC~ (T) ,
( 7.2.6)
where ER is the expectation taken with respect to the Brownian motion on (MR, gR) driven by the same B~, and
HFR,s "-- ER D'R,sF + -~ R,~
~R,taicuRR,t(D'R,tF)dtl jzs
with DR,s, ~R,s,Ric R and Un,t the corresponding quantities defined on (MR, gR). (b) Let PR denote the Riemannian distance function to o on MR. For any m/> 1, let
PR,m(~/) "-- 0~i~2 max TM pR(xiT/2m)
~/ e (MR)To.
We have (see e.g. [104, p.252]) 2m
n'n,~PR,m = ~d ~ ( s
i A Si) U-1 R,~ VnpR,m,
8i "-"
iT 2m,
i--O
where V~ is the gradient operator on (MR, gR) with respect to the i-th component. Since (almost surely) only one of the gradients {V~pR,m}O
ID'R,sPR,ml <. 1,
S E [0, T], m f> 1,
(7.2.7)
oa
0
-I
tO
-~i
0
bO
~ +
~oo
~D
hD
~
~D
9
Oo
/A
~i~.
,--,
~o
-.q
XV
XV
9
/A
m
~I
/A
/A
~
(~I
I,,-,, 9
+
o
~o
9
l'O
~
~
~
~
~~.
9
~1
~
Jr-
x
I
~
I
~
8
/A
m
~
+
~
~
/A
I
~ ~
, ~.
m~ ~~
\V
\V
~
q
~
~i
@
~ol ~
~"
+
~
~o
9
~t
~
/A
~
I~ ,-~
~
8 ~
__
,~
+
"l~l
/,IX
~~"
.
~1
~
--.-~
~1 ~
/A
o
~~
9
~
"~l
/~A
~
~
0
~
~ ~
~OO~ ~
9
9
~ m
u $
~.~
XV
~
9
~
~
,[
0
0
~ ~ ~ ~'~
.
~
~
~ ~ ~ - ~
w ~~
~
"li
~o
.
oo
~
~
~
.
~
~
~
~l
~ ~
~
+
~
~' J
~
~
~ ~~r
.
"
o
~' ,._, ~.,
i~
..
~
~
v
~
8
A
~
~
o
~
~-"
c-t--
.
-~
9
o
~
~
u
:::u
\V
o
II
co
~.
<
|
=~
=
o
322
Chapter 7
Some Infinite Dimensional Models
Therefore, if E F = 0 then E F 2 ~< E[F2(hR o fi)2] +
IIF[I~P(TR1 < T)
$(F, F)(8 + T2eTKI(R)K(R) 2) + [[Fll~(3P(TR~ ~ T) + (8T + T3)r Hence, (7.2.1) holds for the desired a. Finally, since Tk ~ oc as k ~ oc and (7.2.4) holds, for any r > 0 one may find R > R1 > 0 such that
+ T 2) + 3P(TR1 ~ T) ~ r.
r
This means that A~ r o and hence a(r) determined in the theorem is finite for any r > 0. [:3 The following theorem is an alternative to Theorem 7.2.1. Theorem
7.2.2
Let ~ :=
efoKl(p(~))d~K(p(xt))2dt. If E( < ee then
(7.2.1) hold a(r) " - i n f {2 + AT" P ( max E(~l~s ) > 2A) ~< r},
~c[O,T]
r>0.
In particular, (7.2.1) holds for T a(r) .= 2 + ~E~.
Proof. have
If E~ < oc then (7.2.5) holds. For F E J~C~(T)with E F -
IHff[ 2 ~ 2E(ID;F[2I~) + ~ •
0,
we
E(ID,FI2I~) dt
E ( e s K~(P(X~))dUK(p(xt))2l~,~s)dt
1(I
<~ 2 E ( [ D ; F [ 2 [ ~ ) + ~
E(lDtFI2l~)dt
)
E(~I~).
(7.2.10)
Combining (7.2.5) with (7.2.10) and letting ~s "- E ( ~ [ ~ ) , for any A > 0 such that P( sup ~ > 2A) ~< r, we have sE[0,T]
E F 2 <~ EF21{~<2~} +
rllF[[~
ElHYl21{sup~[O,T]~<2),}ds + rIlFl[ 2 ~<(2+ AT)8(F, F)+rIIF[[~.
7.2
Analysis on path spaces over Riemannian manifolds
323
This implies the first assertion. Finally, since ~ is a nonnegative martingale, one has P ( sup ~s>~A) <~ 1 1 sE[0,T]
~ 0 -
~Z~.
Hence the second assertion follows from the first. [:] Now, we intend to deduce explicit curvature conditions from the above two theorems for the weak Poinca% inequality (7.2.1). To this end, we have to estimate either P(TR <<.T) or P(suPs~[O,T]~s > A). A natural w a y t o do this is to compare the radial part of the process with one-dimensional diffusion process. So, we first present the following simple lemma for one-dimensional diffusion processes. L e m m a 7.2.3 Consider the reflecting diffusion process {Pt} generated by 1 d2 d L "- 2 dx 2 + b(x)-~x on [0, co) with po - O, where b E C(O, co) is nonnegative.
If there exists c1 > 0 such that c2 := 2sup x ( b ( x ) - ClX) < 00, x>0
then for any ro > 0 and h(t) := roe -2(~+~~ 1+c2
Ee h(t)p~ <~exp 2(Cl + ro)
(1 --
e-2(cl+r~
t > 0.
(7.2.11)
Moreover, for any A, T > O, tC[0,T]
Proof.
1
2(Cl + ro) (1 - e -2(cl
-- h(T)A 2
(7.2.12)
By It6's formula one has
de h(t)p~ = 2pth(t)eh(t)p~dbt + h(t)e h(t)p~ { 1 + 2b(pt)pt + 2h(t)p2t +
h'(t)
h(t) } dt
<<.2pth(t)eh(t)P~dbt + h(t)(1 + c2)eh(t)P~dt, where and in the sequel bt is one-dimensional Brownian motion. This implies (7.2.11) by a standard argument by using Gronwall lemma. Next, since {eh(T)p2t}t>0 is a submartingale (note that b >~ 0 ), we have P ( max pt >~A)-P( max eh(T)P~ >/ eh(T)A2~ <<e-h(T)A2Eeh(T)P~" tE[0,T]
tE[0,T]
Then (7.2.12) followsfrom (7.2.11).
/
324
Chapter 7
Some Infinite Dimensional Models
By (7.2.3), the Laplacian comparison theorem implies Aft ~< v / ( d - 1 ) g l ( f l ) coth (V/Kl(fl)//(d- 1)fl)
d-1 ~< ~ P
+ v / ( d - 1)Kl(p)
outside the point o and its cut-locus cut(o). On the other hand, by [117] we have 1 dp(xt) = dbt + -~Ap(xt)dt - dL~ + d L t l!
I!
where L~ is an increasing process with support contained in cut(o), and L t is the local time of p(xt) at O. Therefore, letting Pt be the reflecting diffusion process in Lemma 7.2.3 with b(x)
-
-
-~
,x
+ v / ( d - 1)K1 (x) . We have
p(xt) <. ,ot a.s. Thus, Lemma 7.2.3 gives the following estimate of P(rR ~< T). Assume there exists Cl > 0 such that
L e m m a 7.2.4
c2 " - s u p { x v / ( d x>O
1 e_l_2c~T Let 5(T) = ~-~
1 ) K I ( x ) - 2clx 2 } < cr
We have
P(TR <~T) = P ( max p(rt) >1R) <<.ed+c2-5(T) R 2 tE[0,T]
R>0.
1 we have 5(T) = roe_2(cl+ro) T = h(T). Put Let r0 = 2T'
Proof.
b(x) then 2 s u p x ( b ( x ) -
1 ( d-1
ClX) - d -
\
+ v/(d - 1)gl (x))
1 + c2. By Lemma 7.2.3 and the fact that
x>0
p(xt) <. Pt, we arrive at P
( t~[0,T] max p(xt) >1 R ) <" P
te[0,T]
~< exp [(d + c2)ro _ 5(T)R21 < ed+c2_5(T)R2" Cl + r0
[-7
C o r o l l a r y 7.2.5 In the following three cases (8,~,~C1(T)) is closable in L2(# T) and (pT)t~O denotes the associated Ornstein-Uhlenbeck semigroup. (1) /f there exist c > 0 and 5 e (0, 2) such that g ( r ) <~ cr ~ for big r, then (7.2.1) holds for ~(r) - exp[01 + 02(log + r - l ) ~/2]
7.2
325
Analysis on path spaces over Riemannian manifolds
for s o m e 01,02 > 0 and all r > O. Consequently, there exist el, C2 > 0 such that l I P / - ]_tTIIc~--+2 ~ Cl exp[--c2(log + t)2/5], t > O, where I1" []~--,2 is the operator norm from L ~ ( # T) to L2(#T). (2) If there exists c > 0 such that K(r) ~ c2r 2 for big r and 2c2T2e l+cTv/3-1 < 1, then (7.2.1) holds for
-1
=
for some Cl > 0 and all r > O. Consequently, [[pT
--
#T]]2
co--+2 ~
t-1
C2
for some c2 > 0 and all t > O. (3) If K1 is bounded, i.e. the Rieci curvature is bounded below, and if
(1)
K(r) < e cr2 for some c e
0, 2~e
and big r. Then (7.2.1) holds for
O~(7")- C1(1-{-K (c2v/log-+-~r-1)) for some
Cl~C2 > 0.
Proof. By Lemma 7.2.4 we see that EK(p(xt)) 2 < c~ in the specific three cases, and hence ($, 7:)($)) is a Dirichlet form on L2(#T). We now prove the corollary for the three cases respectively. (a) Let K(r) < cr ~ for some c > 0 and 5 e (0, 2). Since (7.2.3) holds also for K in place of K1, and since
sup { x v / ( d - l)g(x)- 2Clx2 } <
CX:)
1
for any Cl > 0, it follows from Lemma 7.2.4 that for any 0 < 2-~e' there exists
c(O) > 0 such that P(~-R <. T) <~ c(O)e -~
R > 0.
(7.2.13)
On the other hand,
RR+1 e-TK(s)/2K(s)ds
~ c'e -TcR~
for some c' > 0 and big R. Combining this and (7.2.13), there exist 00, R0 > 0 such that I
v/P(7"R < T)
/R ~+1 e -TK(s)/2 K(s)
ds ) e O~
326
Chapter 7
P(TR <. T) <. e -2~176
Some Infinite Dimensional Models R~> R0.
Then R + 1 E A~ provided R > R0 and e -2e~
(T(8 + T 2) + 3) ~< r.
Therefore, for a given in Theorem 7.2.1 we have
a(r) < exp[01 + 02(log + r - l )
5/2] --" (~(r)
for some 01,02 > 0 and all r > 0. To finish the proof of (1), it remains to estimate [[pT _ #T][~__. 2 by using Theorem 4.1.3. To do this, let ~(t) " - i n f { r > 0" 0 ( r ) l o g r -1 ~ 2t},
t>0.
Assume that r/(t) is reached by r0 E (0, 1). If r0 > e -vq then c)(r0)v~ ~> ~(r0)log r O 1 -- 2t. Thus
1 ( l o g ( 2 x / t ) - 01)+}2/~ 1 =" r/2 (t). ro 1 ) exp I{ O22 Therefore,
?~($) -- r 0 ~ max {e -x/~, ?~2(t) -1 } ~ Cl exp[-c2(log + t) 2/~] for some Cl, C2 > 0 and all t > 0. Then the desired result follows from IIP - Tll2~--,2 ~< ~(t) according to Theorem 4.1.3. (b) Let K(r) <. c2r 2 for some c > 0 and big r. Then Lemma 7.2.4 applies with 2Cl " - c v / d - 1, i.e., there exists c~ > 0 such that
- P(TR <. T) <<.c'e -5(T)R2 ,
If 2c2T2el+cT ~
< 1 then 5(T) "- ~1e - l - 2 c ~ T ' 2T /5 " - maxse[O,T ] p(Xs), we obtain
> c2T.
E~ ~ TEe c2T~2~4 < (:X). Therefore, the desired result follows from Theorem 7.2.2.
R>0.
Hence, letting
7.2
327
Analysis on path spaces over Riemannian manifolds
1 (c) If K1 is bounded, then (7.2.13) holds for any 0 < ~-~e" If K ( r ) < e cr2 1 for some c < 2~e and big r, then there exists 01 > 0 such that
1
(l+fR+le -TKI(s)/2 )
v/P(7"R ~ T)
JR
g(8)
ds
R~
~ eol
1 (log r - 1 )~1 E A~ for small r ~> 0. Therefore, by Theorem for big R. Thus, Oll 7.2.1, (7.2.1) holds for some a with
oL(r) "-- 8 -~- e TKI (~ K
(1 V/l~ r-1 ) O1
for small r > 0. But one may take a to be decreasing in (7.2.1), then (7.2.1) holds for
oL(r) "--CI(1-+- K(r
r-l))
for some Cl, c2 > 0 and all r > 0. 7.2.2
Weak Poincar6 spaces
[:3
inequality on infinite-time
interval path
Next, we consider the infinite time-interval path space Mo~. In this case Theorems 7.2.1 and 7.2.2 are no longer valid. According to Theorem 7.2.6 below, the weak Poinca% inequality implies very likely that the curvature is less negative than - c p -2 for big p. Let ~ C ~ "- [.JT>0 ~,~C~(T) and let (E ~ , ~ ( E ~ ) ) be a closed extension of (E, ~ C ~ ) in the L2-space with respect to the distribution of the Brownian motion starting at o. T h e o r e m 7.2.6 Let M be connected. If ( 8 c r is irreducible then M is a Liouville manifold, i.e. all bounded harmonic functions are constant. Consequently, if M is a Cartan-Hadamard manifold with sectional curvature satisfying - c l p 2 ~ Sect ~ - c 2 p -2 (7.2.14) for some c1, c2 > 0 and big p, then (E cr ~(8cr Proof.
Let Ft e ~ C ~ with F t ( x . ) " - u ( x t ) , x .
is reducible. e M ~ . We have
F} - E~(x~) 2, 8(F~, F~) - t~lWl2(x~),
t > 0.
(7.2.15)
Since u is harmonic,
EFt - ~(o), fo T ] E l V u i 2 ( x ~ ) d t
-
EU(XT)2 _ ~(o) 2
T > 0.
(7.2.16)
328
Chapter 7
Some Infinite Dimensional Models
Thus, IEu(xt) 2 is increasing in t. Since u is harmonic and bounded, Ft is a bounded continuous martingale. Then by the martingale convergence theorem there exists For such that Ft converges to For a.s. Moreover, by (7.2.15) and (7.2.16) we have
er 8(Ft, Ft) dt t
ElVul2(xt)dt < Ilull~ < ~ .
Then there exists a sequence tn ~ oc such that 8 cr (Ft~, Ft~ ) := 8 ( F t , , Ft~ ) ---* 0 as n --* oc. Thus, For e ~ ( 8 c~) and 8 ~ ( F ~ , F ~ ) - O . But by (3.2) one has EF~ -(EFt)
2-
jr0(X)ZlVul2(xt)dt>
since u is a nonconstant harmonic function. ducible.
0
Therefore, (gooo, ~(ozc~)) is re-
According to Theorem 7.2.6, to prove the weak Poincar6 inequality it would be natural to consider the following condition IRicl ~< K ( p ) " -
c
(1 + p)5
(7.217)
for some c > 0 and some 6 > 2. Under this condition, if p(xt) ~ c~ fast enough as t --~ c~, then the influence of [Ricxtl will be under control in order to get a weak Poincar~ inequality from (7.2.5). To study the escape rate of the Brownian motion, we need a condition on the lower bound of Ap, which leads to the drift term of p(rt).
Assume that o is a pole with A p >~ 0 for some 0 > 5 and P 2 (7.2.17) holds for some 6 > 2 and some c E (0, 0 - 5). Then for any p > 0-1' there exists c(p) > 0 such that
Theorem
7.2.7
"
EF 2 <~c~(r)e(F, F ) § rllFII 2cr
m
r>0,
FE~C~,
holds for a(r) - c(p)r -p. Consequently, lira log IIP~ - #~ t--~ logt
Proof.
c~---.2
0- 1 4 "
(a) Note that
lim
6~2
6(0-- 2,5- 1)
=O-5>c>0
EF-0,
(7.2.18)
+
to
/A
9
I
, . ~
I
/A
+
I>
B.~ i ~_.~
"
"
u
"~
I>
~--
(I)
0~
i...,9
9
Oo
~
I
e~
+
II
~"
~
N
+
~
H,--,
V
~
~~ ' "
9
~
N
~
/A
"" II
[',,.3
t'O
N
N
~ . . ~ ~...~
L~
N
3
~.
~I~
N
~--., ~--.-~ ~'----,
/A
+ 9"~
D.., ~
~" ~ c~ "
/A >"
~
+
"~"
+o~ ~
+
8---~
~
\V >,
/A
+
+
I--~
+
+
8~
\V >,
II
[=z=]
~
~
~
Ou
~,.
,
Iz" 0 0" 0.., ~ rjo
"t~ V ol
~
0
V
:~ %"
m
t ' ~ l:::b
rlh
~,,0
+~
\v
~
"li
~
~
9
I
rl'/
"~ ""
--~
V
I
I
~lb
~
~
I=
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~
IZ
i-i.
0-q
,-.,.
oo
A
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fll
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N
~
tO
o
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o
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4-
+
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Cn
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W
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II
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V
..
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~
I
~-'-,I
V
?
~
9
~"-~ ~
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~
V
+
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~~.~
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I~'
1=
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,~
oa
t,o
t'O
-..4
~
.
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V
9 ~
-',,1
~
W
I
~
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~
~
~
,~
\V
~'~
o
~~
V
~
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I
-I-
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-1"-
--
~
r~
i..~.
W
+
/A
W
O
O
0
- F ~ t,o
13-,
+
W
e,,i,.
13..,
~..
~"
F
o
~
o8
i,-i~
W
e,,t,.
r
+
("b
iJ~ 9
'-"
i-,o
i-,.i
0r'3 O
7.2
Analysis on path spaces over Riemannian manifolds
331
where (1 + 8) ~+~'+1 r/s "-- (I + p(xs)) v
(1 + s) (~-2)/2 (l+p(xs)) v
Letting
1
{
v
a "-- ? sup~>o (1 + s ) r / - -~(0 - v - 1)~7(v+2)/v
}
e (0, cx~)
we see that ~(~> 0) is a supermartingale. Then
(sup ) ~>o (1 + p(x~)) v ~> )~ ~<
~
A>0.
-
Thus
.<
v,(1 + ~),
~>0.
Combining this with (7.2.24), we obtain
1+~
Z F 2 ~< A~/n IIFJI~ + 8 ( F , F )
( 2 + C3 fo~ (1 +
Ads ) si-(-i+e)a/v "
Noting that (1 + ~)6 > 1, there exists c4 > 0 such that v
E F 2 ~< (1 +
a)~X-~/~llFII~ +
(2 + c 4 A ) E ( F , F ) ,
A>0.
This implies (7.2.18) for a(r) - cr-UV for some c > 0 and all r e (0, 1]. Since 5/v < p and since one may take a(r) - 1 in (7.2.18) for r ~> 1, there exists c(p) > 0 such that (7.2.18) holds for a ( r ) " - c(p)r -p. Thus, by Corollary 4.1.5, we obtain [IP~ - # ~ [ 2[o~2 ~< c't- 1/p for some c~ > 0. Since p > 7.2.3
Transportation distance
2 0-1
is arbitrary, we complete the proof.
cost inequality
on path
[-1
s p a c e s w i t h L 2-
Let M be a connected complete Riemannian manifold with O M either empty or convex. Assume that there is a nonnegative constant K such that Ric(X, X) ~> - K [ X [ 2,
X e TM.
(7.2.25)
Then it is well-known that the (reflecting if O M 7~ 0 ) Brownian motion on M is nonexplosive.
332
Chapter 7
Some Infinite Dimensional Models
For fixed p E M and T > 0, let #T denote the distribution of the (reflecting) Brownian motion starting from p before time T. Then/AT is a probability measure on M [~ := {x." [0,T] ~ M } with or-field ~ T induced by cylindrically measurable functions. Since the diffusion process is continuous, ~T has full measure on the path space MoT "- {x. e C([0, T]; M ) ' x o = o} with a-algebra ~ oT "- MoT N ~ For any T > 0, let
pT (x., y. ) " { ~0 T p(xs,Ys)2ds} 1/2
x.,y. E MTo .
Let W T be the corresponding L2-Wasserstein distance. Moreover, for I = { S l , . " ,Sn} with 0 < Sl < "'" < Sn < T, define the distance on M I "- {xi (Xsl,"" ,Xsn) : Xsi e M , 1 ~ i ~ n} by
{n
19I (xI, YI) "--
}
1//2
~-~(8i+1 -- 8i)p(Xsi, Ysi) 2 i=1
Xl, YI E M I 8n+l - T.
Let W / be the corresponding probability distance. For a probability measure u on MoT, let u I denote its projection onto M I. For two probability measures #1, #2 on MoT, define
T
{
}
W2 (#1, #2) "-- sup W~(#II, # / ) " I C (0, T) is finite . We have the following result where only the lower bound of Ric is involved.
Assume (7.2.25). For any nonnegative measurable function f on MTo with # T ( f ) _ 1, we have
T h e o r e m 7.2.8
W T ( f # T , ~T)2 < ~/rT(f~T, ~T)2 < ~2 (egT -- 1 -- K T ) # T ( f log S).
(7.2.26)
To apply Theorem 6.3.3, we first prove a log-Sobolev inequality for cylindrical functions.
Assume (7.2.25). Let f be a cylindrically smooth function depending only on ( x s l , " " ,Xs~),0 < Sl < ... < Sn ~ T. / f # T ( f 2 ) _ 1 then
Lemma
7.2.9
#T(s2log S2) ~<2~
f
9 .
IVjSl
eK(sJ-8i-l)
K
--
ef4~(sj-8i)
1/2
2
d#T'
(7.2.27)
where so : - 0 and V j denotes the gradient with respect to Xsj.
7.2
Analysis on path spaces over Riemannian manifolds
333
Proof. Let Pt be the semigroup of the (reflecting) Brownian motion. By (7.2.25) we have (see Theorem 5.6.1) Pt(~ 2 log ~2) ~< 2( eKt - 1) ptlv~l 2 + (pt~2)log Pt~ 2 (7.2.28) K for any t ) O, ~ e C~(M). Then (7.2.27) holds for n - 1 since in this case #T(f2 log f2) _ P~I (f2 log f2)(p). Assume that (7.2.27) holds for n ~< k for some k ~> 1, it reminds to prove for n = k 4- 1. Let #{Sl""'sn)(dxs:, " " , dxsn) =P(81,p, dxs1)P(82 -- 81,Xsl, dx~,) 999 x P(sk
-
8k-l,
Xsk_l,
dx~k),
where P(t, x, dy) is the transition kernel of the (reflecting) Brownian motion. Note that for fixed y E M k, it follows from (7.2.25) and Theorem 5.6.1 with t - 8k+l -- 8k that V f M f2(y'Xsk+l)P(sk+l - sk, ", dx,k+ 1) (7.2.29) <~2eK(Sk+l-Sk)/2 / M (If I" IVk+lfl)(Y, Xsk+l )P(Sk+l -- Sk, ", dx~k+ 1). By (7.2.28) with t account, we obtain
=
8k, (7.2.27) with n = k, and taking (7.2.29) into
8k+l-
#T (f2 Iog f2) __ fM" d#{Sl' ,s,}fM (f2 log f2)P(sk+1--8k,Xs,,dxs,+l) <<. 2(~K('~+l - ~ ) - 1 ) , T (irk+IX12) + 2 ;M / ~ 1 #{~:,'",~k}(dxs 1, "'" , dx~k) K k fM f2P(Sk+ 1 -- Sk, Xsk, dXsk+l)
•
{/M (k-J-I(,K(sj_Si_l)_eK(sj_si))i/2) }2 ]fl
Y~lvjfl
K
P ( S k + l - - S k , Xsk,dXsk+:)
2--z 9
k+l /
.
(,K(sj_sj_:)_eK(sj_s~))i/,),
(k+l
lvjfl i=1
d# T .
K
j=i
D C o r o l l a r y 7.2.10 In the situation of Lemma 7.2.9, let I = {81,... ,Sn} with 0 < S l < "'" < Sn <. T and let #I denote the projection of #T onto M I. For any Sn+l > 8 n and any function h: (0, T]---, (0, cxD), we have n
pI(f2 log f 2) ~<2 ~
j=l
#i(ivjf[2)~O 0 .
f~j+l h(s)ds osj
~' ds
[Sn+l Js
eK(t-~)h(t)dt,
334
Chapter 7
Proof.
Note that
Some Infinite Dimensional Models
1,2}
j--i
--1
JSi~i eg(sk-S)ds / 8k+l --I JSk Sn+l
fSj~l h(s)ds
ds
h(t)dt
eK(t-~)h(t)dt
,18 Then the desired result follows from Lemma 7.2.9.
K]
Let pt(x., y.) = p(xt, Yt). We have
L e m m a 7.2.11
(#T X #Tl(P2t) < ~1( 3 d -
1) (eKt - 1),
t e [0, T].
Proof.
Let (xt)t~o and (Yt)t>~o be two independent (reflecting) Brownian motions with x0 - Y0 - P. Since OM is either empty or convex, we have (see [117], [186])
dp(xt , yt)
9
1
x/2dbt + -~(Ap(xt,: ")(yt) + Ap(., yt)(xt))dt- dLt, Z
where bt is one-dimensional Brownian motion and Lt is an increasing process. By (7.2.25) and the Laplacian comparison theorem we have 1 ~(Ap(x, .)(y) + Ap(.,
y)(x)) <. ~ / K ( d - 1)coth (~/K(d- 1)p(x, y)) d-1
< p(x, + v / g ( d - 1). Y) Then, by Ito's formula we obtain
dp(xt, yt) 2 <. 2x/~p(xt, yt)dbt + (2d + 2~/K(d 1)p(xt, yt))dt < 2v/2p(xt, yt)dbt + ( 3 d - 1 + Kp(xt, yt)2)dt. -
Since p(x0, y0) = 0, this implies that 1 ( 3 d - 1)(e Kt --1), ]Ep(xt, Yt) 2 ~ --~
Hence the proof is finished.
t>0. D
7.2
335
Analysis on path spaces over Riemannian manifolds
L e m m a 7.2.12 (~T X
Assume (7.2.25). Let ct "- (etKt - 1)/K, we have exp[a(3d - 1)ct/(1 - 4act)] v / 1 - 4act
#T)(eaP(xt'yt)2)
t e [0, T], a e (0, 1/4ct).
Proof.
By (7.2.28) and the additivity of the log-Sobolev inequality, we have
(Pt • Pt)(~ 2 log ~2) E 2ct(Pt • Pt)([VM• for any t > 0,~ C C~(M x M). Since 5.3.2 this implies that
(Pt x Pt)(e ap2) <
2) 4- (Pt • Pt)(~ 2) log(Pt • Pt)(~ 2)
[VMxMP]
2 --
2, according to Corollary
exp[a((Pt x Pt)(p))2/(1 -4act)] V/1 - 4act '
t > 0.
(7.2.30) K]
Then the proof is finished by Lemma 7.2.11.
Proof of Theorem 7.2.8. For I - {si " 1 <~i ~ n} with 0 < 81 < ' ' " < 8n < T, let fI(xs1 , ' ' " , X8 n ) __ # T ( f ] X s l ,''" , Xsn) and let #I be the projection of #T onto M I. It is easy to check that pI is the Riemannian distance on M I with metric (X, Y ) I "- E ( 8 i + l i
- 8i)(Xsi, Ysi)M,
where X~ (resp. Ys~) is the i-th component of X (resp. Y) which is tangent to M{ ~}. Moreover, let VI denote the corresponding gradient operator, for g E C ~ ( M I) one has n
( r i g , V I g } I -- E ( 8 j + I j=l
-- 8 j ) - l I v j g { 2
Thus, by Theorem 6.3.3 and Corollary 7.2.10 with h -
W I ( f I # I , # I ) 2 ~ 2 # I ( f I l o g f I) ~< 2# 7 (f log f)
~08n/8n+1 ds ,18 ds
1, we obtain
eK(t-s)dt (7.2.31)
eK(t-~)dt.
It remains to prove the first inequality in (7.2.26). Since (MoT ,Pc~) T is a Polish space with Borel a-algebra ~oT, where p~T(x., Y.) "--suPt~ [0,T] p(xt, Yt), {#T, f#T} is tight. Moreover, for any compact set D C MTo and any ~ E ~ ( f #T, #T) one has
7r((D • D) r <. #T(Dr + (f #T)(Dr
336
Chapter 7
Then
~ ( f # T #T)
n --+ c~, where 5 ( I n ) : : sk~ < T -
{In}
is tight too. Let
Some Infinite Dimensional Models
be increasing such t h a t
5(In) ~ 0 as
maxl
sk~+l}. For each n ~> 1 let 7rI~ E
~(fI~#I~,#I~)
such t h a t
1
#~((p~)2) < w ~ ( f ~ ~ , , ~ ) 2 + -n. Let
7rn(')"- [ 7rI~(dxi~, dyi~)[(f# T) • #T]('[xI~,YI~), i.e.
for any F C ~ T x
.J
o~T one has
.~(A) .- fM,~ •
[(f'v) • ,V](AIx~, y~)#~ (dx~, dy~).
T h e n {Trn} C W(f#T#T). Let {Trn,} be a subsequence such t h a t 7rn, --+ 7r weakly for a probability measure 7r on M T x M T, then 7r e W(f#T, #T). Thus for any n >~ 1 and any N > 0, let p ~ be defined a s pin but with p replaced by p A N, we have 7r((p~)2) _
lim lr I~' ((p~y)2) nt--+(~
Int 2) < (1 + ~ ) w [ ( f , ~ ,v)~ + (1 + ~-1) sup .~' (Ip~ - PN I ~'>n
(7.Z.32)
]p(x~,y~)- p(xt, Yt)[ <~ p(x~,xt)§ p(y~, yt),
for any ~ > 0. Noting t h a t sup 7rIn' ( ] p ~ --
n'>n
PN [
< 2
NA
sup
O<s
p(Xs, xt)
we have
(f#T+#T)dx,
which converges to zero as n --+ oc according to the d o m i n a t e d convergence theorem. By letting first n ~ oc t h e n N --+ oc and finally s --+ 0 in (7.2.32), we complete the proof. [3 Note t h a t (7.2.26) does not make sense when T --+ ~ . To establish a t r a n s p o r t a t i o n cost inequality which works also for T = c~, we introduce below a modified distance: for K ~> 0, T > 0 and h E C[0, c~) with h(r) > 0 for r > 0 s u c h t h a t
/01
s-lh(s)ds { ~0T
pT(x" Y') "--
< oc, define
h(s)p(xs y~)2 '
fo dr f T h(t)eK(t_~)dt
ds
}1/2
.
Let W T'h be the corresponding L2-Wasserstein distance. Let I~ T'h be defined as IYT with pI replaced by
{~
pI (xI, YI) "--
j=l
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8n+l "- T.
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338
Chapter 7
Some Infinite Dimensional Models
Then
w~,h(f#c~, #c~)2 ~< fMT
(P~)2dTrT •
T
+ 2 f ~ h(s)[p(xs, zs) 2 + P(Ys, zs)2]Tr(dx., dy.) ~ dr fr~176 eK(t-r) h(t)dt ds -/My
(phT)2dTrT •
T
+ Z .~-/~ h(s) f_f:p(x~,~ -f-~zs)[(le_g( 2 7---r f) (dX.)h(t~ #~] as
- fMZ XMoT
+
Combining this with the first assertion, we arrive at
w~,h (f #c~, lzc~)2 < wT, h(fT#T, #T)2 + e(T). Then (7.2.33) follows by noting that lim r T---,c~
- 0 according to (7.2.34).
[:3
R e m a r k 7.2.1 Theorems 7.2.8 and 7.2.13 can be extended to diffusion processes with time-dependent drifts. Consider, for instance, the process generated by L(., t ) " - ~1(A + Zt), where Zt is a Cl-vector field for each t E [0, T) In particular, let Qt(x, y) be the transition density of the Brownian motion and let
Zt "- 2V log aOT_t(., q),
t e [0, T)
for a fixed point q, then the distribution of the diffusion process starting from p is the Brownian bridge measure on the pinned path space {x. E M T 9XT -- q}. Assume that g. e C([0, T); [0, c~)) such that Ric(X, X ) - (VxZt, X) >1-KtlXl 2,
t e [0, T), X e TM.
Then
W T (f#T #T)2 <~I~T (f#T #T)2 <<2#T (f log f)
f0Tds f TeK'(t-~) dt
for all f >~ 0 with #T(f) _ 1. Moreover, Theorem 7.2.13 remains true with K replaced by Kt in the definitions of p~ and p~.
7.2
Analysis on path spaces over Riemannian manifolds
7.2.4
339
T r a n s p o r t a t i o n c o s t i n e q u a l i t y on p a t h s p a c e s w i t h t h e intrinsic distance
Consider the situation in the last subsection. Ff(x.) "- f(xx),x, e MTo . Let
~ C 1 "-- { F f ' f
For any f c CI(MX), let
e C I ( M I) for some p a r t i t i o n / o f [0, T]}.
Recall that the square field operator determined by the HI-derivative reads n
_F(Ff, Fg)(X.) = ~
(si A sj)(xI),
f, g e CI(MI),
i,j--1
where //~,~j is the stochastic parallel transport along the diffusion path x from Tx~ M to Tx~j M. Define the intrinsic distance
pTH(X., y.) "-- sup{IF(x.) - F(y.)I" F Theorem
,To (F
E
o~'C1,/'(F, F) ~ 1}.
Assume that M is compact and let p E [1, 2]. If
7.2.14
- ,To 2-p
<<.C#(F, F) "= CUTo(F(F, F)),
F
C o~'C 1
F~>0
(r.2.35)
then 1)
(F# o Jo ) < I V
'
2-p
F >1 O, #To(F ) -- 1.
In particular, let K "-- sup{]lRicu
-
u-lvz(71"u)II
9u e
O(M)},
where 7r " O(M) ~ M is the projection and ]1" [] is the operator norm on I~d, we have W pT (F#oT, #oT) ~< e KT ~/2#To (F log F),
F ~> 0, #oT (F) - 1.
Proof. For any partition I = {s l, s2,... , Sn} with so := 0 < Sl < "'" < Sn T, and for any z - ( z s l , " " ,Zs~) C M I, define u z ( d x . ) " - #To(dx.]xi -- z). Then for any f C CI(M~), one has E(Ff , FI) "- fM V(Ff , Ff )dpTo
To
n
= I'M ~
or i,j=l
(si A s j ) { / / ~ , , j V ~ f , V~jf)d#oT
n
= fM E I i,j=l
(ai5 ( x I ) v s ~ f ( x I ) ' v s , f(xI)}#Io(dxI)'
(7.2.36)
340
Chapter 7
Some Infinite Dimensional Models
where
l<.i,j<.n.
aiI(xi) := (si A sj) fMT//si,sjdux,,
For a vector field X on M I, let Xi denote its projection on T M {~}. Define n
(AIX)j := ~
a~jXi,
j-l,...
,n.
i--1
Then A I T M I ---, T M I is continuous (even differentiable according to [81]) with AI(xi) a strictly positive definite, symmetric linear operator at each xi. Moreover, by (7.2.36), (7.2.35) implies (Ip) on M I for the measure # / a n d the form
ei(.f, Y) "-
#Io((AIVM, f , VM, f)).
Thus, it follows from Theorem 6.3.3 that
WPPn(FZ#/'#~
~
C(#Zo((FI) 2/p) - 1) ~< P i c ( # T ~ " 2-p 2 p
for all F > 0 with #oT(F) -- 1, where F I ( x i ) " -
(7"2"37)
#To(FJxi ) and
p I ( x I , yI) " - sup{Jf(xI) - f(Yz)l " f e CI(MI),
(7.2.38) Since p I ( x i , y i ) is equivalent to lows form the triangle inequality in (xi,yI). Therefore, given I a n d a sequence {I (m) :-- {0 < _(m)
the Riemannian distance on M I, it foland (7.2.38) that di(xi, yi) is continuous {Sl,... ,Sn} with 0 < Sl < " " < Sn <~ T 8~m) < "'" < 8(nm) ~ T}}m)l such t h a t
maxi Jsi - ~i
J ~ 0 as m ~ c~, we have di(m ) (xI(m), yI(m) ) ~ di(zi, yi) as m ~ oc. Moreover, it is trivial to see from the definition that if I C J then di(xr, YI) <. dj(xa,ya) for any z., y. E M T. Thus, pTH(x., y.) "-- supdI(xI, YI) -- lim dIn (Xln, YI,), I n---,c~
In "-- {i2-nT " 1 <. i <~ 2n}. (7.2.39)
Consequently, pT is measurable. According to (7.2.37) there exists 7rn e c ~ ( F I n ~ I n , #Ion) such that
M In
xMIn
pI~dTrn <. p
~c(#To (F2/P) - 1) 1 ~ . 2 p n -
(7.2.40)
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
341
Let #~(dx., dy.) "= ~ ( d x i ~ , dyln)[(F#To ) • #To](dxi,~, dY~ lxln, Yln). We have ~'n E C~(F#oT , ].toT). Since MoT with metric d(x., y.)"- maxte[0,T] p(xt, Yt) is a Polish space and the metric induces the a-algebra o~oT, the set {/toT, F#oT 9 n >~ 1} is tight so that W(F#oT, poT) is tight too. So, we may assume that #~ ~ 7r weakly as n --, ec. Moreover, we may regard di~ as a continuous (and hence bounded since M is compact) cylindrical function on MoT. Thus, ~ e W(FpTo, #To) and by (7.2.39), (7.2.38) and (7.2.40), WPP/~(F#~
P~
~< /M
oT•
T
pTdTr -- lim /M
= lim
lim [
<~ lim
lim fM
n---~c~
n~cc r n ~ J MT • MT
n ----> (:X) m-----~ ( : ~
Im
oT•
T
dI~ dTr
di~ d#m
x M ~m
p/mdTrm ~ p ~ C ( # T ( F 2 / p ) - I )
2 -- p
Finally, the second assertion follows from the first one and the log-Sobolev inequality in [30, Theorem 1]. Note that we are considering L = A + Z and the time interval [0, T], this is equivalent to consider 1 (A + Z) with time interval [0, 2T], then one should replace e K in [30, Theorem 1] by e 2KT.
E]
7.2.2 Since for any t c [0, T] and any x. c MTo, the function F(y.) "- p(xt, yt)/t satisfies F(F, F) - 1, we conclude that pTH(X., y.) >I
Remark
1
p(xt, yt)/t. Therefore, Theorem 7.2.14 holds also for supt~[O,T]-tP(xt,Yt) in place of the intrinsic distance.
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
Harnack and functional inequalities for the transition semigroup Pt of OrnsteinUhlenbeck processes on infinite dimensional (e.g. separable Hilbert) spaces ]HI, i.e. solution to stochastic differential equations of type
dXt = AXtdt + dWt, where A 9 ]I-]I--~ I[-I[is a linear operator generating a C0-semigroup Tt := etA, t >/ 0, on NI and (Wt)t>~o is an 1HI-valued Brownian motion, have been studied quite
342
Chapter 7
Some Infinite Dimensional Models
intensively in the literature. The fact that Pt may have an invariant measure makes Ornstein-Uhlenbeck processes in many respects better reference processes in infinite dimensions than Brownian motion. If one wants to allow possible jumps for the trajectories (which occur naturally in models described by a partial differential equations perturbed by a pulsating random force), suitable corresponding reference processes are processes solving stochastic differential equations of type
dXt = AXtdt + dYt, where now (Yt)t~>0 is a Levy process on ]HI (i.e. has stationary independent increments and starts at zero), completely determined by a respective negative definite function )~ : H ~ C. Their corresponding transition semigroups Pt are called generalized Mehler semigroups. They as well admit invariant probability measures in many cases and are thus of high interest. So, in case of jumps they might serve as reference operators in infinite dimensions instead of a Laplacian or a local Ornstein-Uhlenbeck operator. Hence, both from a probabilistic and an analytic point of view, there is motivation to analyze generalized Mehler semigroups, especially also in infinite dimensions. This section is devoted to prove Harnack and functional inequalities (as Poincar~ and log-Sobolev inequalities) for such generalized Mehler semigroups. To keep preliminaries to a minimum, from now on we do not refer to infinite dimensional processes solving the above stochastic equations, but define Pt independently just in terms of Tt and a negative definite function )~ 9 ]HI --, C. Let ]HI be a (real) separable Silbert space with inner product (., .), norm I1" II, and Borel a-field ~(]H[). Let (Tt)t>~o be a C0-(i.e., strongly continuous) semigroup of bounded linear operators on ]HI generated by A. We identify ]HI with its dual space. Let (#t)t>>.obe a family of probability measures on ]HI satisfying ~t+~ - (~t o T / ~ ) * #~,
s, t >/O.
Let ~b(H) denote the set of all bounded measurable real-valued functions on ]HI. Then the generalized Mehler semigroup associated with Tt and #t is given by (see [24] for details)
Ptf (x) - / ~ f (Ttx + y)#t(dy),
f
e i~b(]H[), Z e ]HI.
(7.3.1)
It is shown in [24] that, up to some regularity conditions, the characteristic function of #t must have the form /2t(~) := exp
[/0 -
]
A(T*~)ds ,
~ e H , t > 0,
(7.3.2)
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
343
where Ts* denotes the adjoint operator of Ts and A is a negative definite function (or equivalently e -~ is positive definite) on ]HI with A(0) - 0. Let us consider except the last subsection the case where A is Sazonovcontinuous. Recall that a function f is called nuclear if there exist {ak} C I~ CO
and {xk} C H such that
f(x) - Eak(X'Xkl
for all x e ]HI. The Sazonov
k=l
topology is induced by all nuclear functions, see e.g. [180] for details. But the main results can also be applied to the non-Sazonov-continuous case via an approximation procedure as made in w below. By the Sazonov-continuity, A has a unique Levy-Khinchin representation (see e.g. [151, Theorem VI.4.10])
1
A(~) - -i(~, a) + ~(~, R ~ ) -
/H ( ei(~'x)
i<~'x) )K(dx)
1 - 1 + Iix[I2
'
~EH,
(7.3.3) where a E H, R is a symmetric trace class operator on ]HI, and K is a Levy s
measure on ~(]H[), i.e. K({0}) - 0 a n d / ( 1 7.3.1
A [[xll2)K(dx) <
cO.
Some general results
As mentioned above we aim to establish Harnack and functional inequalities for Pt. Since H can be infinite-dimensional, it turns out that the Li-Yau type Harnack inequality involving dimensions does not work. Therefore, we follow the line of [192] (see also w to establish a dimension-free narnack inequality. To this end, we exploit an explicit formula for the underlying square field operators obtained recently in [129, Proposition 4.1]. Thus, we make use of the following assumptions made in [128], [129]. (A7.3.1)
Pt
has an invariant probability measure #.
(A7.3.2) There exists (Xn)n>~l C H consisting of eigenvectors of A* and separating the points of ]HI. Assumption (A7.3.2) holds if, in particular, A is self-adjoint with compact resolvent. Assumption (A7.3.1) holds provided the following (stronger) condition holds (see [90, Theorem 3.1]) (A'7.3.1)
(i)
Ttx ---+0 as t ~ oo for
all x e ]HI;
supt>otr( fotT~RT:ds) < c~; (iii) f o ds f~(1 A []T~x]]2)K(dx) < c~;
(ii)
(iv) a ~ "- lim { f0 t Tsads+ ~ot/Mds
t~
exists.
Tsx ( 1 + I[T~xll 1 2 -i + 1I[x]]2 ~/ K ( d x ) ~
344
Chapter 7
Some Infinite Dimensional Models
In this case Pt has a unique invariant probability measure # with characteristic function/5(~) := e - ~ ( ~ ) , where , ~ is given by (7.3.3) with a ~ , R ~ "-
/0
TsRT2ds and K ~ "=
/0
K o T~-lds replacing a, R and K respectively.
Indeed, for any f E Cb(IHI) and any x E IH[ one has Ptf(x) ~ # ( f ) as t ~ c~ (of. the proof of [89, Theorem 3.11). Let
p(x, y) .- { i n f { l l z l l
"z e
(R1/2)-l(x- y ) } ,
if x - y E R1/2][']I, otherwise.
In fact, there exists an alternative definition of p(x, y). Let G be the orthogonal complement of KerR 1/2, and let S "= R1/2IG 9 G ~ R 1 / 2 G - " I[-Io. Then S is a continuous isomorphism (see [129]) and (H0, (., ")~o) is a Hilbert space with (v, W)]HIo "-- ( S - i v , S - l w ) for v, w E H0. It is clear that p(x, y) - I I x -YlIMo "= . \1/2
<x - y, x - Y/H0 if (x - y) E 1HI0. We shall use the following two conditions. (A7.3.3) that
TtRN C R1/2]I-]I and there is a strictly positive hi E C[0, oo) such IITtRxlIHo <<.v/hl(t)IIRxllM0,
x E N, t ) 0.
(A7.3.4) There is a strictly positive h2 E C[0, cc) such that K o Tt - 1 h2(t)K, t ~> 0. Let us recall the formula for the square field operator obtained in [129]. Let W be the space of functions f of the form .f(x) -- F ( ( ~ I , x > , ' "
, <~m, x>),
x E ]HI
for some m >~ 1 and F E S(N m) "= {F E CC~(Nm)'suPxERm m
m
IxZD~F(x)l <
oo, c~, fl E Z~}, where x ~ "- H xjZr , D ~ "= H 0~ j Then W is dense in LP(#) j=l
j=l
for any p >f 1 and is a core of N(L), the L2(#)-domain of the generator L of Pt, see [128]. Moreover, according t o [129, Proposition 4.1], for any f E W,
F(f, f ) " - L f a - 2 f L f - ( R D f , D f) + s If(.) - f(. + y)]2K(dy),
(7.3.4)
where D f denotes the Fr~chet derivative of f. Note that by [128, Theorem 1.1 (i)], F(f, f ) E Cb(lHI) for f E W.
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
345
Assume (A7.3.2) and (A7.3.3). Then
T h e o r e m 7.3.1
]Ptf(x)l 2 <. [Ptf2(y)] exp
[
~(x'~) ~ ] fthV~s~_-ld s , du
---
f E Cb(N),x,y E H,t > O.
\ - /
(7.3.5) If in particular K = O, then for any p > 1 and any f E Cb(IH[),
IPtf(x)l p <~ [Ptlflp(y)] exp [ L..
pp(x,~t,_;,y)e 1' 2(p- 1)jo.118)-148
x, y ~ M, t >
O.
...J
(7.3.6) To prove Theorem 7.3.1, we need the following two lemmas. (1) If (A7.3.3) holds and PtW C W for all t > 0, then
L e m m a 7.3.2
v / ( R D P t f , DPtf} <. v/hi (t)Pt v / ( R D f , D f),
t >. 0, f E W.
(7.3.7)
(2) If (A7.3.4) holds, then for any t ~ 0, f E W and x E H,
/ I.,,Ix+,>-
{/I,I.
}Ix>. (7.3.s)
Proof. Let f C W, t > 0. It is easy to see that ]]Dfl ] is bounded, hence because Ptf c W we conclude by Lebesgue's dominated convergence theorem that for all x, z E ]I-]I (DPt f (x),
z) -/(Df(T~x + y), Ttz)pt(dy)
- Pt((Df(.), Ttz))(x).
Hence, since by assumption TtR]H[ C R1/2H, we obtain that
(Rz, DPtf} <~Pt(IIRi/2DfI] 9 ]IS-1TtRzI]) ~ v/hl(t)(z, R z ) P t v / ( R D f , Df). Choosing z := D P t f ( x ) , x C ]HI, the first assertion follows. Next, we have
~[Ptf(x + y) - Ptf(x)]2K(dy) -
/.{/.
[f(Ttx + Try + z) - f(Ttx + z)]#t(dz)
J H xlI-II
Hence
(7.3.8)
}'
K(dy)
[f(Ttx + y + z) - f(Ttx + z)]2#t(dz)(g o Tt-1)(dy)
holds.
E]
346
Chapter 7
Lemma7.3.3 ]HI • ]HI ~ (0, oo)
Some Infinite Dimensional Models
If there exists p E [1, cxD) and a measurable function ~ : such that
IPtf(x)l p <<.[PtlflP(y)] ~(x, y),
x, y e H, f e ~b(]I-][).
(7.3.9)
Then Pt has transition density pt(x, y) with respect to # satisfying II~t(x, ")lip, -<
{/H ~(x, #(dy)}-lip y) ,
x E M,
(7.3.10)
where p~ "= P . p-1 Proof. Let x E ]HI be fixed. For any A E ~(]I-]I) with #(A) = 0, (7.3.9) yields that
(#t o OTtO(A))p < [PtlA(y)]4;'(x,y). Then
(#t o OTtl(A)) p/H ~(z,y) #(dy) <~~ [Pt 1A (y)]#(dy) = #(A) = 0. Therefore, PtlA(x) ----#t o 0Tt 1x (A) - 0 and hence #t o 0 T- 1t x is absolutely "continuous with respect to #. Let Qt(x, .) denote the Radon-Nikodym derivative. Dividing (7.3.9) by ~(x, y) and integrating with respect to #(dy) we thus obtain by (A7.3.1) for all f E ~b(H)
(~t(x, .), f)i=(~,) <~ Ilfllp Thus, ]]Qt(x, )lip'< { / s o
I/M #(dy)]-i/p ~ixlY)
"
~(x,y)l#(dy)}-l/P --
,
[-7
Proof of Theorem 7.3.1. We first prove (7.3.5) for nonnegative f E W. For E (0, 1), let P(~) be the generalized gehler semigroup determined by Tt and )~e, where )~ is given by (7.3.3) with K replaced by K~(dx) "-
l{a:e<~llall
We note that for all results from [128] used below the assumption (A7.3.1) (although stated in [128, Theorem 1.3]) was not used in their proofs. By [128, Theorem 1.3 ( i ) a n d Proposition 3.3] one has Pt,eW C W for any t 1> 0. For x # y with p(x, y) < oo, by (7.3.7)
(Dp(~) (p(~) f)2, y _ x) =
inf
zE(nl/2)-l(x-y)
(Dp(~) (p(c) f)2, R1/2 z)
p(x, y)v/(RDP(:) (P(6) f) 2, DP(:) (P(~)f) 2) ~< 2p(x,
y)V/hl(t - s)P(: ) { (P(e) f)v/(RDP (~)f, DP (~)f) }.
(7.3.11)
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
347
Let
(y -- X) fo hi ( t - u)-ldu fo hi ( u ) - l d u
xs "= x + r
" - l o g P(9(P(e)f + e)2(x,),
, ~ [0, t].
It follows from [128, Theorem 1.1 (ii)], (7.3.4) and (7.3.11) that 1
d hi (t - s)
/0
hi (u) - l d u
<~
hi ( t - 8) 1/2 ~
p(x,
~<
y)2-
hi (t - 8)(fo hi (~t)-ldu) 2'
hi( u ) - l d u
, e [0, t],
where g
o~
(RDP(e) f, DP(~) f) (p(e) f + ~)2
Therefore, we obtain
r
< r
p(x,y) 2 fo hi (u)-ldu"
Hence
(p(e)f + ~)2(y) <~ iN(e)(f + ~.)2(x)] exp fo hi (u)-ldu " According to [128, Corollary 3.5], one has Pt(e)f ~ Ptf uniformly as e ~ 0 for any f E W, therefore (7.3.a) follows by letting e ~ 0. For general f E W and any n ~> 1, let hn E C ~ ( R ) such that 0 <~ hn(r) <~
Irl, hn(r) - 0 for Irl ~< 1, and hn(r) - Irl for Ir[ ) 2. Then hn(f) E W. n n Applying (7.3.5) for hn(f) in place of f then letting n ~ c~, we arrive at
P(x'Y)~ 1 (p~lfl)~(y) < [p~f~(x)]~xp fo <(ui :ld~
348
Chapter 7
Some Infinite Dimensional Models
This implies (7.3.5) for f E W. We are now going to prove (7.3.5) for f e Cb(H). Let ]HIn "-- span{x1,... , Xn}, 7Tn" ]HI ----> ~-~n is the orthogonal projection, n ~> 1. For f E Cb(H) and n ~> 1, there exists gn E Cb(]Rn) such that fn "-- f o 7rn -- gn((Xl, "),''" , (Xn, ")).
Let {gnm }m>~l C C ~ (R n) be uniformly bounded such that gnm ~ gn pointwisely as m --, oc. By (7.3.5) for fnm "= gnm((Xl, " ) , ' ' " , (Xn, ")) and using the dominated convergence theorem, we obtain
IPtfn(X)I 2 -
lim
m----+ (:x3
IPtfn~(x)l 2
<. { lira
Ptlfnml2(y)} ~(x, y)
- [Pt[fn[2(y)] ~ ( x , y),
[ p(x,y) 2 ] where ~(x, y ) ' - exp fo h11isi - l d s " Since f E Cb(H) and 7rnX ~ x as n ---, c~ for any x c ]I-]I,using the dominated convergence theorem once again, we prove (7.3.5) for f e Cb(H). If M = 0, then L is a second order differential operator, hence (7.3.6) follows by repeating the above argument with := log P(~_)(P(e)f + e)P(xs)
r
K]
and using the chain rule. Corollary that
7.3.4
Assume (A7.3.1), (A7.3.2) and (A7.3.3). Assume further lt~l/2 Tt]I-]I C ~t ]H[,
where Rt "=
/o
TsRT~ds. Let I1
lip
t > O,
(7.3.12)
denote the norm in LP(#). Then:
(1) Pt is strong Feller, in particular, Pt(x, dy) has a density Or(X, y) with respect to # for x E supp# (where as usual supp# is the smallest closed set in H whose complement has #-measure zero). Furthermore, (7.3.5) (resp. (7.3.6) if M = O) holds for all f e ~b(H) and one has
II~t(x,')ll2 ~ {/Mexp
p(x,y)2
[-
]
fohl(s) "lds #(dy)
}-1/2 ,
x E supp#, t > O,
(7.3.13)
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
349
1 where as usual we set -0 "- co, so the right-hand side is equal to oc if #(x + leo) - O. If in particular K - O, then for any p, p' > 1 with p-1 + p,-1 _ 1, ,,Qt(x, "),,p, ~ { / M e x p [ -
pp(x'Y)2 2 ( p - 1)fo "hlis) - l d s
] ,(dy)
}-l/p ,
x e ]HI,t > 0.
(7.3.14) (2) If there exists t > 0 such that TtR1/2H is dense in ]HI, then supp# - ]HI
and p is the unique invariant probability measure of Pt. Proof. (a) By Theorem 7.3.1 and the proof of Lemma 7.3.3, it suffices to prove that Pt is strong Feller. Let pl and pt2 be probability measures with Fourier transforms ftl(~) "- exp
[/0 t[i(Ts~ , a) - -~(r*~, 1 RT*~)]ds ] ,
/2t2(~) " - e x p
ds
e ~(T;~'z> - 1 - i( ~ ~,x) 1 + I[x[[2 K(dx) .
Let p1 be the generalized Mehler semigroup determined by Tt and pl. If (7.3.12) holds then according to [228] (see also [67]) p1 is strong Feller. For any f e ,.~b(IH[), let
ft(x) "- .f~ f ( x + z)p2t(dz),
x e H.
One has ft E ~b(NI) and
Ptf(x) - f
JH xN
f ( T t x + y + z)#l(dy)p2t(dz) - Pr ft(x),
so, Pt is strong Feller, too. In particular, Pt(x, dy) has a density Or(x, y) with respect to # for all x c supp #. (b) Since TtR1/2H is dense in ]HI for some t > 0, to prove supp# - IE by [10, Proposition 2.7] it suffices to check that # o 0x I is absolutely continuous with respect to # for all x E TtR1/2H. Let x - TtR1/2x '. It follows from (7.3.5) that for any A e ~ ( H ) with #(A) - O, one has for ~(x, y ) ' -
exp[p(x,y)/~othl(s)-lds]
that
fH(PtlA)2( R1/2x' + y)~(R1/2x ' + y , y ) - l p ( d y ) <~ #(PtlA) But 4)(R1/2x ' + y , y ) - i > 0 since R1/2x ' + y - y 0 #-a.e. Thus,
#(A) - O.
E R1/2H, so (PtlA)oOR~/2 x, =
p o O;I(A) - p(Pt(1A o Ox)) -- #((PtlA) o OR1/2xr ) -- O.
350
Chapter 7
Some Infinite Dimensional Models
This means that # o ~x 1 is absolutely continuous with respect to #. Therefore, s u p p # - H. Let #1 and #2 be two invariant probability measures of Pt, then they have full support. Now it is easy to see that they are equivalent. Indeed, if e.g. # I ( B ) - 0 then # I ( P t l B ) -- 0. But PtlB is continuous and #1 has full support, then PtlB ---- 0 so that #2(B) - # 2 ( P t l s ) = 0. Let #2 - - r and 1 2r # " - ~(#1 + #2). We have #2 - 1 + r # -" r Then Pt*r r where Pt* is the adjoint of Pt in L2(#). Since Pt* is sub-Markovian, for any c > 0 one has r
Since #(P~(c - r
- #((c- r
#-a.e. Thus, if # ( ( c - r
- p/c)+
-
r
it follows that" P t * ( c - r
r
> 0 then # ( ( c - r
- (c- r
is an invariant probability
measure for Pt and hence is equivalent to # as explained above. Therefore (c- r > 0 #-a.e. This means that for any c > 0 with #(r < c) > 0, one has #(r < c) = 1. Hence r has to be constant, i.e. ~ 1 ~- # 2 . R e m a r k 7.3.1 (i) It has been proved in [228] (see also [67]) that when M - 0, (7.3.12) is equivalent to the strong Feller property of Pt. For examples where (7.3.12) holds if ]HI is infinite dimensional we refer to [67]. (ii) By definition H0 corresponds to the Cameron-Martin space of the Gaussian part Of (Yt)t~>0 (or equivalently A), so if dimH - c~, then H0 c ]E and p(x, y) - c~ for x - y E H\Ho. As a consequence, Theorem 7.3.1 and Corollary 7.3.4 tell us nothing e.g. in the pure jump case where R - 0. So, our results in Theorem 7.3.1 and Corollary 7.3.4 are of a perturbative type, more precisely, we can only handle perturbations of the Gaussian case. Theorem 7.3.1 and Corollary 7.3.4 are very useful for the analysis of Pt, when the invariant measure # is supported by ]HI0 for which there are plenty of examples in infinite dimensions (cf. the last subsection below). In such a case the analysis of Pt can be done entirely on H0. This is due to the fact that Tt is regularizing in these situations. As another application of our estimate for the square field operator, we have the following Poincar6 and log-Sobolev inequalities. Theorem
7.3.5
Assume (A7.3.1), (A7.3.2), ( A 7 . 3 . 3 ) a n d (A7.3.4). Let
hi " - 0 if R -O, and let C(t) "- ( i t hl(s)ds) V ( i t h2(s)ds). Then Ptf 2 - (Ptf) 2 <<.C(t)PtF(f, f),
t > 0, f E W.
(7.3.15)
0
6h
c~
I
I
v ~
b~
b~
b
A
<..
\V
\V
~
+
~
"
V
~" "~ ~
~
~
~
0 a"
~
II
~
I
II
9
~""
o
e-t'-
-.,,1 ~
I
~-
0~
b~
.,'2",
I
~' ."7",
v
~ ~
~ ~
~
V
%
~-~ ~
~
C~
~
~~176
9
~
c~
~
6h
~
~
+
v
~
o ~
v
b
A
~
~
/A
I~
to
0~
.~.
+
I
v
~-
XV
~
II
~_A
o
r.n
~
~
|
.
co
r
9
rn
.
~
~
v
r~
o
o
0 "0
~ ~
i,.~~
~
~
~
ii
i ~
D"
-~="~
~
o~
.~
H
L'~
~o
r~
V
(~
o
0
~
~
~
0
i..,,~.
c~
r
b~
Or~
c~
i-,~o e-,l--
i.,~o
---
-~
352
Chapter 7
Some Infinite Dimensional Models
This implies (7.3.16) by first integrating over s from 0 to t and then letting ~ --~ 0. K] When K ~ 0 it is not clear how one can prove inequality (7.3.16). In applications one can often check the following condition which is a stronger variant of (M7.3.1). (A~7.3.1) (i), (ii) and (iii) in (A'7.3.1) hold and the limit in (iv) exists with g(dx) replaced by g ~ ( d x ) ' - l{a:e~
Assume that (A'7.3.1), (A7.3.2),
(A7.3.3) and (A7.3.4)
hold.
(1) If C ( oc ) < oo, then
f
#(f2) _ #(f)2 <
w.
If moreover (Ae7.3.1) holds, then # ( ( P t f - #(f))2) ~< e-2t/c(oo)p((f2 _ #(f))2),
t ~> 0, f E L2(#). (7.3.19)
In particular if M = O, then
#(f2 log f2)~< (2 f0 ~ hi (t)dt) #(/'(f, f)), (2) If K -
f E W, # ( f 2 ) _
1.
0 and there exists t > 0 such that [[R-1/2TD1/2 cr t-Lee [[2 := ~t < i,
then
p(f21ogf2)~<
(2
1-at
hi (s)ds) #(F(f, f)),
f E W, # ( f 2 ) _ 1.
Consequently, if in addition (7.3.12) holds, then Pt is compact in LP(#) for all t > 0 and hence the LP-essential spectrum of L is empty. Proof. Since the assumptions imply that P t f ~ # ( f ) as t ~ c~ for any f E Cb (]HI), the Poinca% and log-Sobolev inequalities in (1) immediately follow from Theorem 7.3.5. Let a~ ) and K ~ ) be given in (A'7.3.1) (iii) and (iv) with K replaced by K~, and let A~) be defined by (7.3.3) for a~ ), R~ and K ~ ) replacing a, R and K respectively. Then P(~) has a unique invariant probability measure #(~) with characteristic function ft(~)(~) "- e- ~)(~) It is clear that/2(~)(~) ~ ft(~) for all ~ E ]HI and hence #(~) --~ # weakly as ~ --~ 0. By letting t ~ c~ in (7.3.18), we obtain
#(~)(f2) _ #(~)(f)2 ~< C(oc)#(~)(F(~)(f, f)),
f
w.
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
353
Since, according to [128], P(~)W C W C ~(L(~)), where L (~) is the generator of p(e) in L2(p(e)), this Poincar6 inequality implies that -,
(e)(f))2) ~< #(e)((f _ # (~)(f))2)e-2t/c(or
for all t > 0 and all f E W. Letting c ~ 0 we prove (7.3.19) for all f c W and hence for all f E L2(#) since W is dense in L2(#). So, let us prove (2). Since st < 1, according to [59], [89] we have [[Pt[Ip--~q <~ 1 for any p > 1 and q - 1+ ( p - 1)ct 1. By (7.3.16) and Proposition 7.3.7 below, we arrive at p(f2 log f2) ~<
1 -2p ct ~o t h ( s ) d s ) p ( F ( f , f ) ) ,
f E W,p(f2)-1.
The proof is then finished by letting p ~ 1. Finally, by Corollary 7.3.4 if (7.3.12) holds then Pt(x, dy) has a density with respect to # for any t > 0, x E supp #. Therefore, Pt is compact in LP(#) for any t > 0 according to Theorem 3.1.7. Hence the essential spectrum of L in LP(#) is empty, see e.g. [114, Theorem 6.29]. [:] P r o p o s i t i o n 7.3.7 If there exist t > 0, 1 < p < q < oo and C1, 62 > 0 such that [[Pt[]p~q ~< 61 and
pt(f2 log f2) ~< C 2 P t r ( f , f ) + P t f 2 log P t f 2,
fcW,
then #(f2 log f2) ~< C2p(q - 1) p ( F ( f , f)) + pq q-p q-p Proof.
log c1,
f E W, p ( f 2 ) _ 1.
See the proof of Theorem 5.7.1.
D
Finally, we formulate a result on the Harnack inequality relying on conditions on #t and Tt only without referring to R and K. T h e o r e m 7.3.8 Let t > O. If for some x C H the measure #t o OTt1 is absolutely continuous with respect to #t, where Ox(y) := x + y , y E H, we define
71t(x , z) 9 d#t o 0T -1X(z), dpt -
z M,
and 71t(x, . ) " - oc otherwise. Furthermore, for x E H and p' E (1, oc] we set (7.3.20)
354
Chapter 7
Some Infinite Dimensional Models
Then
IP~f(x)l ~ < P~lfl~(y)~,~,(x- y)~,
f e ~b(H), x, y e H,
(7,3,21)
where p E [1, c~) such that p-1 4_ pl-1 _ 1. Finally, set
M C(t, p', e) "-where as usual c~ 1
0 and -~ 1
#(dx) [f ~t,p' (x - y)-P#(dy)] 1+~
c~. Then we have
IIP~IIp-~(~+~> < c(t,p',~) 1/(1+~>. Proof.
(7.3,22)
(7,3.23)
For any bounded, nonnegative measurable f, we have
Ptf (x) - / / ( T ~ x + z).~(dz) - / S(T,y + z).~ o 0 -1~(~_~)(dz) iiJ
< / f ( T t y + z)~t(x - y, z)#t(dz) < [PtfP(y)] I/p ~t,p,(X - y). This proves (7.3.21). Next, let f be such that #(Ifl p) - 1, by (7.3.21)we have
,P~f (~),. J ~.~.(x - y)-P#(dy)
<~ 1.
Then (7.3.23) follows from (7.3.22)immediately.
Yl
Proposition 7.3.9
Consider the situation of Theorem 7.3.8. If C ( t , p ~, O) < oo for some t > 0 and p' E (1, oo], then Ps is compact on LP(#) for any s > t. Proof.
By (7.3.21), for any f with I l f l l p - 1, we have 1
IPtf(x)lP <<"f ~ , p , ( X - y)-p#(dy)" If C(t,p',O) < oo, then ~ "- {Ptf'llfllp <~ 1} is uniformly integrable in LP(#). Moreover, by Lemma 7.3.3 Pt has density with respect to #, hence as shown in the proof of Corollary 7.3.6 it follows that P~ is compact in LP(#) for s > t. D
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
7.3.2
355
Some examples
We first present an example to illustrate Theorem 7.3.1 and Corollary 7.3.6. E x a m p l e 7.3.1 Let a and R be arbitrary as in (7.3.3) and let Tt - e - Z t I for some/3 > 0. (1) If g is such that (A'7.3.1) (iii), (iv) hold, i.e.
fo~ dt
f~[IA (llxll2e-2~t)]K(dx) 1 1 +e-2~tllxll2
e-Ztx
<
1 + IIx
112)K(dx) e ]HI exits.
Then (7.3.5) holds for h i ( t ) " - e -zt. (2) Let P(d~) be a finite measure on $1 := {~ e ][-]I : I[~11 -- 1} and let r e (0, 1). Write x := (~, s) if x = s~, s > 0. Define K by K(dx) "-- l ( o , c c ) ( s ) s - ( l + r ) P ( d ~ ) d s . Then (A7.3.4) holds for h2(t) - e -~zt. following Poinca% inequality holds p ( f 2 ) _ #(f)2 ~
1
Hence (A'7.3.1) is satisfied and the
p(r(f, f)),
W.
(7.3.24)
f E W, # ( f 2 ) _ 1,
(7.3.25)
f e
(3) Let M = 0, then p(f2 log f 2 ) ~ C p ( F ( f , f)), holds for C = 2//9. Since T t x ~ 0 as t ~ c~, (A'7.3.1) holds. (A7.3.2) is trivial since H is separable and each ~ c IHI is an eigenvector of Tt. Therefore, Theorem 7.3.1 applies and hence (1) follows. Next, (3) follows directly from Corollary 7.3.6. Finally, it is easy to see that M given in (2) is a Levy measure satisfying (A'7.3.1) (iii), (iv), and for any A e ~(H) Proof.
K o T t 1 (A) -
P(d~) 1
/0
lezt A(S~) s-(1 +r) ds e-r~tlA(s~)s-(l+r)ds
-- e-rC~tZ(A).
Hence (A7.3.4) holds for h2(t) - e - r z t . Thus, the Poinca% inequality follows from Corollary 7.3.6. [:3
356
Chapter 7
Some Infinite Dimensional Models
In the next example Pt is given by a Gaussian part and an a-stable jump part. We show that this semigroup possesses the Poincar~ inequality but is not hyperbounded. E x a m p l e 7.3.2 Let IHI- IRn, Ttx - e-t#x and A(~) -[1~[[ a + 6[[~[[2, where /3 > 0, 5 ~> 0, a E (0, 2) are constants. Then: (1) For any t > 0, there exists C(t) > 0 such that
IP~Y(x)l <. P~lYl(y)C(t)(1 + e-(n+~)~llx- Ylln+~), (2) For ~ny t > 0 ~nd ~ny 1 < p < q < ~ ,
IIP~II~-~ -
s >
t.
~.
(3) The Poincar~ inequality (7.3.24) holds with r replaced by 1 A a. - - 0 then it holds with r replaced by a.
Proof.
If
Obviously, /~t(~) -- exp
By Theorem
1
- ~--~(1 - e -
~#t
)11,'11
-~(5
1 - e -2zt) 11r
3.1 in [90], the unique invariant measure # is determined by ~(~)
-
exp
-
-
1 ~11r
, 2] . ~ll~ll
C~
Let t > 0 be fixed and ~1, ~2 two probability measures with /)1(~) -- exp [
Then #t
I1r a# (1 - e-aZt) ]
d/]l . /]1 * ~2. Let Pl (X) -- ~dx
1
CI(1 + Ilxll n+~)
~2(~)
-
exp
-
6 2(1 ~11~:11
e-2Zt)],
Since ~1 is a-stable, we have (see [20]) C1 Pl (x)
i + []x[]n+a
for some C1 > 1 and any x E IR". Since ~2 is a normal distribution for 5 > 0 and a Dirac measure for 5 - 0 , it is easy to check that 1 d#t C2 C2(1 + I[xl["+~) ~< ~lt(x)"- dx ~< 1 + ][x[["+~
(7.3.26)
for all x and some constant C2 > 1. Therefore, there exists C3 > 1 such that sup [~t(x, z)[ ~< C3(1 + [[x[[n+a). Z
Obviously, C3 = C3(t) can be taken such that C ( t ) " Hence the first assertion follows from Theorem 7.3.8.
SUps~>t C3(s) < oo.
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
357
Next, it is well-known that for the present A one has K ( d x ) - c(a, n) x ]lxll-(~+~)dx for some constant c(a, n) > 0. Then (A7.3.4) holds for h2(t) e -z~t, and hence the desired Poincard inequality follows from Corollary 7.3.6. Finally, let us fix t > 0 and 1 < p < q < oc. Take c C (a/q, a/p) and put f(x) = ]lxl]e. It is easy to see that Ilfllp < oc since the same type of estimate in (7.3.26) holds for r j ( x ) " - d# dx" But by (7.3.26)
p
f(x) -
f
II - Zx + YlI
(Y)dY/>
1 f I,e-tZx+yll~dy 1 + IlYlln+
~> ~ 1 f{lyll~
#i[Ptf] q) ) 9[{"
xlI~>Ch//C4}
C6(C4llxlle - Ch)q 1 + Ilxll n+':'
dx -
oe
since eq > c~. This means that IIPtllp-~q = oc. D We now intend to construct a non-Gaussian generalized Mehler semigroup which is hyperbounded in L 1 and hence in L p for all p ~> 1. Although the semigroup below looks like a perturbation of a symmetric Gaussian generalized Mehler semigroup, the Ll-hyperboundedness can not be obtained by perturbation arguments. The reason is that the Gaussian generalized Mehler semigroup itself is not L l-hyperbounded with respect to its invariant measure. Indeed, for a symmetric semigroup the L l-hyperboundedness is equivalent to the ultraboundedness. But the Gaussian generalized Mehler semigroup is not ultrabounded. Example
7.3.3
Let ]I-]I - ]t~n. Consider
Ttx - e-tZx
and A(~) - 51I~ll2 +
n
2zE i=1 1 - 2'/3'5 + ~
> 0. Then (Tt,$) determines a generalized Mehler semi-
group Pt satisfying IPtf(x)l
~< Ptlfl(y)(1
-
e-213t)-(n+l)/2 exp
[C + 2(n - 1)/3t +
e-tZlx -
YI1]
(7.3.27) for all f ~ ~ b ( R n) and some C -
C(n, 5,/3)
> 0, where Ixll -
Ixil.
~
Hence
i
lion(x, )11~ ~< (1 - e-2/~t) -n/2 exp[C +
e-Ztlxll],
t > 0, x E R n. (7.3.28)
Consequently, IIP~llp--,p(l+~) < oc for any p >/ 1, t > 0 and e ~ (0, e tz - 1). Therefore, Pt is compact in LP(#) and the LP-essential spectrum of L is empty for a n y t > 0 a n d p ~ > l .
358
Chapter 7
Some Infinite Dimensional Models
Moreover, if 6 - 0 then Pt is not uniformly integrable in LP(#) for any p E [1, oo) and any t > O, hence is not hyperbounded. 1 Proof. Obviously, f i ( ~ ) " - 1 4 - ~ is positive definite since it is the characteristie function of (1~ e - X~ldxi ) • (6o )n-1 , where 60 denotes the Dirae measure at 0 on I~. Then 1 +r r
is negative definite for all i, and hence so
- 1 - fi(r
is A(~). We have ~t(~) "-- exp
exp exp x
[/o --
[_
/
(1
A(e-ZS~)ds
e-2~t)~llr
-
]
- exp [ -
e-2/~t)~[[~[[ 2
(1
23
L
n
2
~I 1 + e-2/~t~:2 1 4-r
i=1
1 - e -2#t
2# i=1 [_ (1 e-2#t)~llr ]
)
-
e-2(n-k)Zt(l _ e-2#t)k
e -2nc~t +
}
k=l il ,... ,ik=l Let /~(r . - lim f t t ( r t--+oc
exp[-6I[r (I + r ... (I + r
'
then # is the unique invariant measure of Pt. Let
r
-- V/2Zr6(1 _ e_2#t)
/3r 2 ] 26(1 - e -2/~t) ' r
exp [-
1
--
~e-
Irl
We have d#t
1
(271") n
~t(x) "- dx
JfR e-i(~'x) n
/~t (~)d~
--
e -2nzt
n
n
r
i--1 n
4- ~ e-2(n-k)/~t(1 - e-2#t) k k=l n il,...,ik=l
k . j=l
II
r (xi).
i~il,...,ik
Noting that /3r 2 6(1 e-2# ~) 26(1 - e-2# ~) ~< - r + 2/3
--rH---
2#'
(Xi)
rER.
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
we conclude t h a t there exists
C1 - C1 (n, 5,/3) > 0 such that
e-2(n-1)C~t V/1 _ e-2/3te-lX 1-C1 ~ Moreover, there is C -
359
7it(X) ~
(1 -- e - 2 Z t ) - n / 2 e - Xll+C1 9 (7.3.29)
dp C ( n , 5,/3) > 0 such that e -]xll-C ~ rl(x) "-- d x
e - z l l + C . Thus, (7.3.28) follows. Furthermore, (7.3.29)yields that z)
.-
-
~< (1
T
x)
-e-2/3t) -(n+l)/2 exp[e-C~t]xll -t- 2C1 + 2 ( n - 1)r
X,Z E N n.
This implies (7.3.27) by Theorem 7.3.8. Next, there exist C2, C3, C4 > 0 depending also on t such that
e_2Cl f
~t,~, ( d(xy )- y) ) 0'2
exp
[ -
e
-tZlx
-
y 11 ] , ( d y )
i> c3 dR[n exp [-- e-t/3lx -- Yll - - l y ] l ] d y ) C4 exp [ - e-t~lxll ].
Therefore, C(t, (:x),E) < ~ ( l + e ) fN n exp [(1 + e ) e - t Z l x [ 1 ] p ( d x ) < oc whenever (1 + e)e -tz < 1. Then it follows from Theorem 7.3.8 that IlPt[ll__~l+~ < o~, For a n y p > 1, let q - ( l + e ) p f o r oo, Pl -- 1, ql -- 1 + r T h e n 1 -
r =
t
P
Pl
c < e t g - 1. e < et ~ - l .
1-r
1
~
-
,
P0
=
q
Let r -
(7.3.30)
1/p, po - q o -
r 1-r ~ + ~ . ql q0
Thus, ]]Ptl]p~p(l+c) < oc for any p ~> 1, t > 0 and any e E (0, e tz - 1) according to (7.3.30), the Riesz-Thorin interpolation theorem and L e m m a 7.3.3. Finally, if 5 - 0 then
~t -- e-2n/3t(60)n Jr- e(~, t)#, where n
n
-
k=lil,'",ik=l (I Jr-~i21)''" (i -~-~i2k) ' #(d~) "- 2 -n I I exp
i=1
- ~
i=1
Ixi] d x i .
360
Chapter 7
Some Infinite Dimensional Models
Thus, for f i> 0 one has
P t f (x) >~ e-2nZt f (e-Ztx) =" P t f (x). The proof is then finished by noting that Pt is not uniformly integrable in LP(#) for any p e [1, c~) and any t > 0. [i] Finally, to see that the log-Sobolev constants given in Corollary 7.3.6 can be sharp, let Ttx - e-t/2x. We have hi(t) - st - e -t. Then each of (1) and (2) in Corollary 7.3.6 implies (7.3.25) for C = 2. This constant is sharp as is well known for the standard Ornstein-Uhlenbeck process. We remark that one may also prove the log-Sobolev inequality by using a curvature condition. Let A be the generator of Tt, then Pt is generated by
L f (x) "- ~tr(RD2 f (x)) + (D f (x), Ax). One defines the curvature operator by 1
F2(f , f ) "- -~Lr(f , f ) - F ( f , L f), where F ( f , g) := ( R D f , D f). We say that the curvature of L has lower bound K E R , if F2(f, f ) >~ g F ( f , f )
(7.3.31)
for all f E ~r where ~2' is a core of L, stable by L and Pt and by the action of composition with C cr real functions which are zero at zero. According to Theorem 5.6.1, (7.3.31) is equivalent to (7.3.7) with hi(t) replaced by e -2Kt, therefore we prove (7.3.25) for
C "- 2
e_2Ktd t _
fOX)
K 1V--O"
(7.3.32)
Intuitively, if hi(t) is not an exponential function, then Corollary 7.3.6 could provide a sharper constant than (7.3.32). To see this, we present a simple example below, in which we get a better constant (but we do not know the best). Consider ]HI - I~2, R - I, Ttx - (e-txl -+- cte-tx2, e-tx2), c+c 2 c ~> 0. Then Corollary 7.3.6 (1) implies (7.3.25) for C = 1 + ~ , while 2 2 c (7.3.32) gives C = ( 2 - c) + which is larger than 1 + ~ for c > 1.
Example
7.3.4
7.3
Functional and Harnack inequalities for generalized Mehler semigroups
361
We note that Tt is a semigroup since
Proof.
T t T s x - (e - t (e - s x i -~- cse - s x 2) -+- cte - r e - s x2 , e -(s+t) x2 ) -- Ts+tx.
Obviously,
c2?)llxll 2.
IIT~xll 2 - e-2t[(xl + ctx2) 2 + x 2] <. e-2t(1 -I- ct + Then Corollary 7.3.6 (1)implies (7.3.25) for C.-2fo
c -+- c 2
cr e -2t (1 + ct + c2t2)dt - 1 +
2
On the other hand, we have 1A_(xl_cx2)
L-~
0
~XXl
0
1
X2ox---~ "= ~A + x .
By the Bochner formula, (7.3.3!) is equivalent to
-
. - yl ~
0
0
+ Y2 O X 2 '
y E
R2 .
Noting that - ( V y X , Y } - (yl -cy2)y~ + y~, we see that the best choice of K is c So, the proof is completed. V]
1-~.
7.3.3
A generalized Mehler semig'roup associated Dirichlet heat semigroup
with the
We consider the following model discussed in [129] stimulated by the study of stochastic heat equations. Consider ]HI "- L2((0, 1); dx) and let Tt be the Dirichlet heat semigroup of A on (0, 1). Let
~(~)--II~ll ~ + I1~11~,
~ c M,
where a E (0, 2) is fixed. Let #t be defined through its Fourier transform /~t(~) "- exp
[/0 -
]
(IIT~II 2 + IIT~ll~)ds.
(7.3.33)
It was shown in [129] that/2t is Sazonov continuous for all t > 0, so #t indeed exists as a probability measure on (]HI,~(H)). Furthermore, it was shown in [129] that the generalized Mehler semigroup Pt determined by A and Tt has an invariant probability measure # "- p ~ . Let {~n} be the set of unit eigenfunctions of the Dirichlet Laplacian on (0, 1), i.e. we have A ~ n -- -nTr2~n, n >~ 1. Observing that the above A is indeed not Sazonov continuous, we use finite-dimensional approximations to apply our results. For n ~> 1, let Hn "s p a n { ~ l , . . . , ~n}. Let 7rn" H ~ ]~n be the natural projection.
362
Chapter 7
Theorem
Some Infinite Dimensional Models
Let Pt be determined by A and Tt above. Then
7.3.10
[Ptf(x)] 2 <. [Ptf2(y)] exp [ e~2t - 1
'
f E Cb(H), t > 0, x, y e N.
'
(7.3.34)
Next, for any t > 0 and any 1 < p < q < oc, one has
[[Ptllp~q
-
oo.
Proof.
Let #p denote the projection of #t on Nn. Since {~n}n)l are eigenvectors of Tt, one has TtlH[, - Nn for any t ~> 0, n ~> 1. Let f E Cb(IH[) with f = f o 7rn, one has, for all x E ]I-]I
Ptf (x) = ~ f (Ttx + y)#t(dy) - ~ n f (TtTrnX + y)#~(dy) =: P(n) f (rrnX). Since [[Tt[[ ~< e -~2t, t ~> 0, it follows from Theorem 7.3.1 (with 1HI,Pt replaced by IH[n,Pt(n) respectively) that
[Ptf (x)] 2
F
(P(n)f(TrnX))2 < [P(n)f2(Trny)] exp
<~ [Ptf 2(y)] exp
e~ 2 t - 1
'
71-2117~n(X -- y)II e 7r2t - 1
2]
x, y C H, t > 0.
Thus, (7.3.34) holds. For general f E Cb(N), first applying (7.3.34) to fn " f o 7rn then letting n ~ c~, we prove (7.3.34). To see that Pt is not hyperbounded, let t > 0 and 1 < p < q < c~ be fixed. Since for f with f = f o 7rl one has P t f - ( p ( 1 ) f ) o 7rl, we have [[Pt]lp~q )
[Ipt~l)[[Lp(,(1))~Lq(t,(1)),--where #(1) stands for the projection of # on HI. But according to Example 7.3.2 (2) with n = 1, we have [IP(1)][Lp(,(1))~Lq(,(1)) (:x:).
7.4
[:]
Notes
w is essentially due to [219] and the other parts of w is taken from [162]. It is well-known that the following Poincar~ inequality holds (see [216, Remark
1.4]): 7r"(F2) ~ < / r dTr. ~ ( D x F ) 2 # ( d x ) + 7r.(F) 2,
F E L2(Tr.).
See [18], [118], [220] for extensions to a class of Gibbs measures with E = N d. Thus, in our present setting one has gap(oZf)f> 1 provided 0 - 1 and In := qn,n+l >f 1, q,,m = 0 for m > n + 1. On the other hand, by Theorems 7.1.6
7.4
Notes
363
and 1.3.11, g a p ( S f ) > 0 if and only if infn~>0 nln > 0. Therefore, Theorem 7.1.6 provides a much weaker (and sharp) condition for gap(g~ > 0. According to Theorem 7.1.6, 8 f does not satisfy the super Poincar(~ inequality. From the proof one sees that the reason is that we do not allow the particles to move. Thus, to deduce stronger functional inequalities, one has to enlarge the Dirichlet form so that the particles move intensively enough on the based space E. In this case explicit conditions have been presented in [162] for the enlarged Dirchilet form to satisfy the super Poincar~, the log-Sobolev, the super log-Sobolev inequalities as well as the inequalities studied in Chapter 6. An immediate question is how can one extended these results to infiniteconfiguration spaces. In the latter case the intensity measure # is (a-finite) infinite, so that the Poisson measure 7ru has full mass on OO
" xi E E,i >~ 1}.
-
i--1
In this case, a jump process on F ~ will not be able to clearly related to any Markov chain on Z+ as we did for the finite intensity case, since the total number of particles is always infinite. But one may also modify our construction of ozf to consider e.g. (:x:)
(DxF)2#~(dx),
8 f (F, F ) " - / F + n=l
n
where {ln } is a sequence of nonnegative numbers and #n a proper measure on E n for n ~> 1. Indeed, when l l - - l , In - 0 for n > 1 and/-tl - - #, this Dirichlet form reduces to the birth-death one
For this special ease the Poinear@ inequality (and even more general ~5-entropy inequality) already existed, see [216]. More precisely, let ~5 E C2(0, cx~) such that 9 " ~> 0 and @(s, t ) " - ~(s + t) - ~(s) - ~'(s) is convex in (s, t).
(7.4.1)
Then the proof of [216, Theorem 1.1] implies that, for a a-finite measure #, EatS, " - 7 r ~ ( ~ ( F ) ) -
~(Tr~(F))~< .fv dTr~ .[p (Dx~(F)
<<. ]~ drct, ./~ (Dz~(F))(Dx~'(F))p(dx),
F
~1(F)Dx F) p(dx)
I
E
Ll(Tr#), F > O. (7.4.2)
364
Chapter 7
Some Infinite Dimensional Models
In particular, the inequality E n t ~ ~<
f~ dTr~/E (D~~(F))(Dx ~'(F))#(dx),
F E Ll(Tr~), F > 0
(7.4.3) is equivalent to the exponential convergence of the semigroup associated to 8b~ in the ~-entropy Ent~,, see [32] and the references within. If ~(r) "- r log r, then (7.4.3) reduces to the so-called modified (or L 1-) log-Sobolev inequality, see [15]. This inequality is equivalent to the usual (L 2-) log-Sobolev inequality in the diffusion case, but is strictly weaker in the jump case. For instance, the proof of Theorem 7.1.6 (3) implies that OZb~does not satisfy the super Poincar~ inequality, but by (7.4.3) there holds the modified log-Sobolev inequality. The Poincar~ inequality was also established recently for some Gibbs measures with a nice pair-potential, see [18], [118], [220] for details. In terms of our present study, one may hope to extend these inequalities (e.g. (7.4.2)) to more general jump processes o n / ' ~ . Indeed, even in the case where # is finite, the inequality (7.4.2)or (7.4.3) is still to be studied. Of course, if ~(r) := for p > 1, our argument in the proof of Theorem 7.1.6 (1) also implies that the ~5-entropy inequality
rp
Ent~0,, (F) ~< C ~
(~(F)(~)-
~(F)(~)-
~'(F)(F)(F(rl)- F(')')))zro,,(dq,)q(~,drl)
xF
LI(zro,~),F
holds for some C > 0 and all F E > 0, if and only if the ~-entrop inequality holds for the corresponding Q-process on Z+. But this argument does not apply to e.g. ~(r) := r log r. Therefore, the equivalence of the modified log-Sobolev inequality for the Q-process and that for the induced jump process on F is still to be verified. w and w are based on [208], For the finite-time interval case, it is not clear whether the weak Poincar~ inequality holds as soon as the Brownian motion is nonexplosive. Theorem 7.2.6 is an improvement of a result by Aida [1] which says that Poincar~ inequality on M ~ implies the Liouville property of M. He also considered compact manifolds with non-Liouville Riemannian covers. Let M be compact and let M be a Riemannian cover of M. Let be the lift of the Brownian motion 7t to ~r with ~ - o. For f E C~(~r), by [1, Lemma 4.1] one has F E ~ ( 8 ) , where/~t (~/)"- f ( ~ ) , and oz(/ht,/St) - tElVf[2(~),
t > 0.
Thus, the proof of Theorem 7.2.6 leads to the following result.
7.4
Notes
365
P r o p o s i t i o n 7.4.1 Let M be a compact connected Riemannian manifold. If M has a non-Liouville Riemannian cover, then (oz ~ , 2(oz~ is reducible. From this result we see that the converse of the first assertion in Theorem 7.2.6 is not true, i.e. the Liouville property of M is not enough to imply the irreducibility of (d~176~(oz)). Indeed, let M be a compact connected Riemannian manifold with sectional curvature pinched by two negative numbers, then its universal cover is simply connected with the same curvature property and hence has non-trivial bounded harmonic functions according to [11]. Therefore, by Proposition 7.4.1 the Dirichlet form (8 ~ , ~ ( E ~ ) ) is reducible. But M is a Liouville manifold because of the compactness. Such a Riemannian manifold possesses some special analysis properties, see e.g. [170] for a lower bound depending on the volume (rather than the diameter as usual) of the first eigenvalue. The transportation cost inequality with the intrinsic distance was first proved by [92] on the flat path space (see also [87]). The corresponding result in w is an extension of Gentil's result to the Riemannin setting. The L 2distance as well as the uniform distance were also considered by [75] and [221] for the transportation cost inequality on the path space of a diffusion process on ]Rd. Very recently, the Talagrand inequality was established by [86] on the path and loop groups using the Cameron-Martin distance. w is based on [161]. Generalized Mehler semigroups, initiated in [24], have been studied intenaively in the past seven years, see [24], [58], [70], [89], [128], [129] and references within. In particular, in [128] the corresponding generators have been identified as pseudo-differentiM operators on ]HI with explicit symbols. The study in this section is stimulated by [192] where the dimension-free Harnack inequality was established for diffusion processes on manifolds, see also w The inequality (7.3.21) is an easy direct consequence of the explicit expression of Pt as already realized by Kusuoka (see page 270 in [120]) for the classical Mehler formula, i.e. in a Gaussian case. For the model considered in w the question whether the Poincar~ inequality holds is still open. Let us indicate the connection of Theorems 7.3.10 and 7.3.1. We can "make" A Sazonov continuous by extending it to a larger space ]HI1 such that 1HI C 1HI1 is Hilbert-Schmidt. This produces in a standard way a positive definite trace class operator R on H1 such that H - R1/2]H[1, so H takes the role of H0 and 1H[1 the role of H above. Then the Levy process corresponding to A lives on ]H[1, but its invariant measure # on H0. We refer to [129] for more details. So, one can also apply Theorem 7.3.1 to prove Theorem 7.3.10.
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Index (E, ~ , p): measure space, 2 (L, ~(L)): linear operator, 1 (g~, ~(g~)): bilinear form, 2 //s,t: stochastic parallel transportation, 92 C(E): the set of all continuous real
to y, 54 R~: resolvent, 2 TM: the vector bundle on M, 13
TxM: the tangent space at x, 12 ~: Banach space, 1 ~ + (E): the set of nonnegative mea-
functions on E, 6 C~-compatible, 12 CP(M): the set of all CP-smooth real functions on M, 12 C~(M): the set of all CP-smooth real functions on M with compact supports, 12 C~(x): the set of C~-smooth real functions defined in a neighborhood of x, 12 C0(E): the set of all continuous real functions with compact supports, 7 Cb(E): the set of all bounded continuous real functions on E, 6 c ~ := ~ + c ~ , 40 H 2,1 (#), 20
surable functions on E, 5 ~b (E): the set of all bounded measurable real functions on E, 6 AO(M): the horizontal Laplace operator, 54 ~ ( 8 ) : the extended domain, 4 E-Cauchy sequence, 4 E-exceptional set, 6 C-nest, 6 8-quasi-continuous function, 6 ~0(c~,/3), 39 ~i (c~,#), 38 ~ 2 ( a , r), 38 "
-
5
F(TM): the set of all vector fields on M, 13 FP(TM): the set of alls CP-vector
Hz: horizontal lift of Z, 54 I(X,X): the index form of X, 17
~ ( E ) : the set of all probability mea-
L*: the adjoint operator of L, 3, 8 L2(p): the set of all #-square-integr-
sures on E, 5 P~: the distribution of a Markov
able real functions, 2 LP(#) : the real LP-space of p, 10 L~ (p)'the complexLP-space of #, 10 M ~ : space of continuous pathes on M starting from x, 55
fields on M, 13
process starting from u, 5 a(V): log-Sobolev constant of A + VV, 193 cut(x): the cut-locus of x, 14 gap(L): the spectral gap of L, 176
N-function, 186
Lg : the g~-Lipschitz constant, 21
Pt or {Pt}t>.o: semigroup, 1
#o: the area measure induced by
Px,y: parallel transportation along
#, 16 p(L): the resolvent set of E, 7
the minimal geodesic from x
Index
377
~ ( E ) : the set of all measurable real functions on E, 2 ~(p, u): the set of all couplings of # and ~, 233 ~(#, u): the set of couplings of # and u, 189 ~ : the curvature operator, 14 at(L): the continuous spectrum of L, 7
ap(L): t h e point spectrum of L, 7 at(L):
the residual spectrum of
L, 7 a(L): the spectrum of L, 7 ad(L):
the discrete spectrum of
L, 9 aess (L): the essential spectrum of L, 9 {E~}: the spectral family or the resolution of the identity, 8
{~'t}t/>0:
filtration, 5 iM: the injectivity radius of M, 14 rnx,y" the mirror reflection operator, 54 C closure, 1 of form, 4 of operator, 1 compact function, 45, 207 condition weak sector, 2 connection, 13 Levi-Civita, 13 constant F-Lipschitz, 21 isoperimetric, 25 convex boundary, 60 convex domain, 60 coordinate neighborhood, 12
coupling, 41 by reflection, 44 Kendall-Cranston's, 53, 55 of diffusion process, 41 Otto-Villani's, 235 successful, 97 curvature, 14 Ricci, 15 sectional, 15 cut-locus, 14 D
differential structure, 12 distance Riemannian, 14 Wasserstein, 189 domain, 60 regular, 67 E
eigenvalue, 62 mixed, 36 principal, 20 the first closed, 62 the first Neumann, 67 essential spectral radius, 10 exponential map, 13 F
field Borel a-, 5 Jacobi, 15 vector, 13 filtration, 5 naturnal, 5 right continuous natural, 5 form, 2 regular Dirichlet, 7 bilinear, 2 closable, 3 closed, 2
378
Index coercive closed, 2 conservative Dirichlet, 3 Dirichlet, 3 positive definite, 2 quasi-regular Dirichlet, 7 second fundamental, 60 symmetric, 2 formula Bochner-WeitzenbSck,. 15 first variational, 16 Green, 16 second variational, 16 Friedrichs extension, 8 G generator, 1 geodesic, 13 minimal, 14 H
Hessian tensor, 15 Horizontal Laplace operator, 54 horizontal lift, 54 I
inequality F-Sobolev, 115 discrete Hardy, 34 FKG, 63 log-Sobolev, 115 Nash, 118 Poincar~, 19 Sobolev, 119 super Poincar~, 115 transportation, 233 weak Poincar~, 145 weighted Hardy, 36 injectivity radius, 14 J
jump process, 22
L Lemma index, 17 lifetime, 5 local (coordinate)chart, 12 log-Sobolev constant, 193 M
manifold, 12 Cartan-Hadamard, 17 differential, 12 Riemannian, 13 Markov process, 5 strong, 6 time-homogeneous, 5 Markov property, 5 measure, 2 Pt-supermedian, 6 of non-semicompactness, 11 of noncompactness, 10 volume, 13 mirror reflection, 54 N
normal frame, 15 O operator, 1 AM-compact, 11 adjoint, 8 asymptotic kernel, 108 bounded, 10 closable, 1 closed, 1 compact, 9 conservative Dirichlet, 3 Dirichlet, 3 divergence, 15 ergodic, 173 essentially self-adjoint, 8 gradient, 15
Functional Inequalities Markov Semigroups and Spectral Theory - Elsevier
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Index
379
horizontal Laplace, 54 kernel, 107 Laplace, 15 linear, 1 Markov, 3 order bounded, 11 positivity-preserving, 173 self-adjoint, 8 sub-Markov, 3 symmetric, 8 orthonormal frame bundle, 53 P
path space, 5, 254 canonical, 5 Poisson measure, 245 Poisson space, 245 positivity improving property, 173 principal eigenvalue, 20 process birth-death, 28 Hunt, 7 jump, 25
Q quasi-left continuous, 7 R
resolution of the identity, 8 resolvent, 2 resolvent set, 7 Ric: the Ricci curvature, 15 S
Sazonov continuous, 274 Sect: the sectional curvature, 15 semigroup, 1 equicontinous, 10 generalized Mehler, 273 hyperbounded, 122 positivity-preserving, 113
strongly continuous (Co), 1 sub-Markov, 3 ultrabounded, 122 special standard process, 6 spectral family, 8 spectral gap, 20 spectral radius, 10 spectrum, 7 continuous, 7 discrete, 9 essential, 9 point, 7 residual, 7 stochastic parallel transportation, 92 stopping time, 6 T tangent space, 12 theorem Bourbaki, 12 Hessian comparison, 17 Hille-Yoshida, 1 Riesz-Thorin's interpolation, 11 spectral mapping, 9 Stein's interpolation, 11 transportation parallel, 13 transportation cost, 233 U uniform norm, 6 uniformly positivity improving property, 165 V vector bundle, 13 W weak spectral gap property, 145 Weyl's criterion, 9
