Pseudo-Differential Equations & Stochastics Over Non-Archimedean Fields (Pure and Applied Mathematics)

(K). It is clear that z is locally constant and (in view of the asymptotics (2.25)), z(x) ->• 0 for ||x|| ->• cxo, z E £V(K). The equality (21 + \}z = y> means that :(a;-y) - z(x)]dy + A^(a;),

x € K.

(2.32)

If z(x) takes negative values at certain points, then (since z(x) -> 0 as ||o;|| —> oo) there exists a point XQ of the global minimum for z(x), and Z(XQ) < 0. Substituting x = XQ in (2.32), we find that the integral in the right-hand side is non-positive and \Z(XQ) < 0 whereas tp(x0) > 0, and we come to a contradiction. •

2.6

Coordinate representation

Let K be a field of zero characteristic. Then K is an extension of the field Qp of a finite degree n. Suppose that n > 2. A rank zero character on the additive group of K can be given by the formula X(x)

= xP(Tr(/rt)),

(2.33)

where Tr : K -> Qp is the trace in K over Qp, d is an exponent of the different (see Chapter 1). Consider a bilinear form

p(x, y) = Tr(xyp-d) ,

x, y € K,

(2.34)

58

Chapter 2

with values in (Q)p. Since p is symmetric and non-degenerate, the field K possesses a p-orthogonal basis ei, . . . ,en over Qp (see [6]): p(ei,ej)=0

if i ^ j ,

p(e»,e») = ^ ^ 0, i = 1, . . . ,n.

If a; = xiei + • • • + xnen , y = ytei +••• + ynen , xt, yt e Qp then

The normalized additive Haar measure on AT can be represented as dx = cjfdxi • • -dxn,

(2.35)

where dxj is the normalized additive Haar measure on Qp, 1

= I J\\xiei+---

Let 21 be a pseudo-differential operator studied in Sect. 2.5. Consider an operator B over Q™ defined by the relation (Bg)(xi,... ,xn) = (2t/)(xiei + --- + xnen) where g € 2?(Qp); / € 2?(-^) is such that

,... ,xn = It is easy to see that B is a pseudo-differential operator F~1Mi>F with the symbol

= 4-

1 a

j=il l^'lp y

fc=i

/

Here F is the n-dimensional Fourier transform i,... ,^ n ) = / Xp(^i^i + - - - + a r n ^ n ) y > ( a r i , . . . ,a; n )dii ...dxn

The basis {ej, ... , en} is called a fiaszs of coordinate representation for the operator 21. Below such a basis will be frequently used as follows. Starting from a pseudo-differential operator A over Q™ with a symbol i ) " - > £ n ) l p > where h is a p-adic quadratic form, a > 0, we find an

Fundamental solutions

59

extension K , a basis of coordinate representation, an appropriate symbol a(£), £ £ K, and a matrix T of order n over Qp such that

where b has the form (2.36). Now A = WflBWT , (WTg)(x) = g(T'x), x £ Qp.The hypersingular integral representation (2.20) implies the representation

= t \\y\r'1G(uy)[g(x1-(T'Y)1,...,xn-(T'Y)n) JK ..,xn}]dy,

(2.37)

where Y = (t/i, . . . , ?/„) € Q™ is determined from the expansion y = y\e\ + • • • + 2/n e n, y £ K. The relation between a and v and the choice of K and a will be specified separately for different n and various classes of forms h. The expression (2.37) makes sense, for example, for locally constant functions g(xi,... ,xn) growing at infinity not faster than max |a:j|JJ~ e , l<j
£ > 0 (or, as above, an integral growth restriction can be imposed). If E(x) is a fundamental solution for the operator 21 constructed in Proposition 2.7, then a fundamental solution S (x\ ,..., xn) for the operator A (understood in just the same way as in Sect. 2.5) is given by the formula

£(Xl , . . . , xn) = | det T\~1E ((T'-lX)iei + ••• + (T'^X)^} ,

(2.38)

where X = (xi, . . .xn). This formula is verified by a straightforward calculation. It is clear that a similar connection is valid for the Green functions, that is fundamental solutions of "shifted" operators 21 + A and A + A. The relations between the operators A and 21 including the formula (2.38) carry over to the case when K is a (non-commutative) division algebra. Namely, let K be a central division algebra over Qp , dim K = m2 (see [115, 156]). One should only substitute m — 1 for d, and the reduced trace t for Tr in (2.33) and (2.34). The hypersingular integral representation (2.20) for general symbols is possible only for a commutative K. However, for the operator DVK the whole reasoning carries over to non-commutative algebras, for example, the quaternion algebra over Qp. The role of the basis of coordinate representation is, of course, in expressing the factor x(£x) 'm the definition of the Fourier transform on K, in terms of the corresponding factor for the multi-dimensional Fourier transform over Qp. Sometimes it is convenient to start from another basis having

60

Chapter 2

in mind a possibility to transform it by a linear transformation into a basis

of coordinate representation, with subsequent use of the formula (2.38). As a first example of this approach, we shall consider the case, where the symbol of A is not defined via a quadratic form (this is postponed until the next section), but equals max I&L

,

a>0.

(2.39)

In a moment it will become clear that the best choice of an extension K, in order to deal with the symbol (2.39), is the unramified extension of the degree n.

Lemma 2.1. Let K be an unramified extension of the field Qp. Then 7-T •. (TD , inhprp wnere nn — — ((J\ \lpj,^

and £j € Qp are the coefficients a canonical basis.

of the expansion of x € K with respect to

Proof. For the unramified extension /3 = p, q — pn. Let x = pNz, \\z\\ = 1, so that ||a;j| = q~N = p~nN . On the other hand, H-ll —n-N\\\\ _ ||_||l/ni| || ll-Z-llmax — P ||*||max — ||*|| ||Z||max>

and it remains to show that ||0||max = 1. Consider the expansion of the element z with respect to the canonical basis {/,-}:

Since the basis is fundamental, we have |^|p < 1 for all j. Since ||/j|| = 1, j = 1, . . . , n, we find that

whence ||z||max = 1. • It follows from Lemma 2.1 that the pseudo-differential operator A with the symbol (2.39) can be studied by means of the reduction to the operator DK™, where K is the unramified extension of Qp of the degree n. For example, it follows immediately from Proposition 2.8 that the Green function of A is continuous outside the origin (and at the origin, if a > n), non-negative, and tends to zero at infinity.

Fundamental solutions

2.7

61

Fundamental solutions of elliptic operators

Let h(£i, . . . ,£„) be a quadratic form on Q™, p ^ 2. We shall consider a pseudo-differential operator A with the symbol |/i(£i, . . . , £n)|p% & > 0. Theorem 2.3. //

fc(6, • • • ,60 5* 0 tofcen |6 |p + - . - + |^|p ^ 0,

(2.40)

i/zere exists a fundamental solution £(x\,... ,xn) for the equation Az = f , which is locally integrable, continuous for (£1,... ,£ n ) 7^ 0, and has at most a power-like growth at infinity. If f € T> (Q™ ) then the function z = £ * / satisfies the equation Az — f , where A is understood as a hyper-singular integral operator (2.37). The proof given below consists of an explicit construction of £ for all possible quadratic forms satisfying the ellipticity condition (2.40). We shall also write down the explicit forms of the hypersingular integral representations (2.37).

a) Binary forms Let /i(£i,£2) satisfy (2.40). Then up to a linear isomorphism

where r € Qp is not a square in Qp (see Sect. 1.4). To obtain all nonequivalent forms one should take a complete set of representatives T of cosets from Qp/Qp2 (here Q* is the multiplicative group of the field Qp,and Q*2 is the subgroup of all squares):

T = £, \e\p = 1 ; T = p ;

T = ep.

In each of the three cases we set K = Q P (-\/T). The absolute value of an element x = xi + \fr x^ € K is given by \\X\\ =

XT —TX. sip-

The operator of multiplication by x = x\ + \fr x
2

2T

XiJ

matrix \X I

1 with respect to the basis {l,\ir}. Thus Trz = 1x\. F I 'V J

Let us consider three possible cases separately. Let T = e. Then K is unramified over Qp whence q = p2,d = 0. We can take e\ = I,e 2 = \fz as a basis of coordinate representation.

62

Chapter 2

It is easy to calculate that c = 1,/ii = 2,/z 2 = 2e, |/fi| p = [^2 p = 1Taking z/ = a, 21 = Z3^-,T = ( nnu

I we obtain a hypersingular integral

representation

,2:2) Q?

A fundamental solution is given by

where for every x £ K,x ^ 0,

-log Hall, .2p 2 logp

if a = 1.

Let r = p. Then if is a totally ramified extension of Qp, q — p, d — 1. One may set /3 = ^/p and take

as a basis of coordinate representation. Then m = -4, /x2 = 4, |/ui |p = |M2|P = 1.

Since K is totally ramified, we have a fundamental basis {1,^/p} (see Sect. 1.3). It follows that the basis {ei,e2} is also fundamental. Indeed, [|ei|| = ||e2|| = 1, and if ||xiei +a;2e2|| < 1, a;i,o;2 £ Qp, that is

then |xi + a^lp < 1, |a;2 — x\ p < 1, whence xi p < 1, |a;2|p < 1. Therefore c = 1. As before we take v = a, 21 = £>£. In this case T = 1/2 f j,

f

Fundamental solutions

63

The expression for a fundamental solution is

£(xi,x2) = E((xi + z 2 )(l - Jp) + (x2 - rci)(l 4- VP)) = E(2x2 - 2 where

{

i _ n-a 0 1 —— - , ifa^l, ! ——-j-HxIl ill ,,

i f a = l,

2; € K,x 7^0. The third case (T = ep) is virtually identical to the second one.

b) Ternary forms If the quadratic form /i(£ii£2>£s) satisfies (2.40) then up to a linear isomorphism

M6, 6, 6) = peiCi2 + £2$ + £3&2

(2-41)

where |ei|p = |e2|P = MP = 1, £2$ +^£3 7^ 0 if 16 IP + I6|p 7^ 0. So it is sufficient to consider only the forms (2.41). Let K be the unramified extension of Qp with degree 3. Its residue field ¥q = O/P consisting of q = p3 elements is an extension of a simple finite field ¥p = TLpjpLp. Let A be a primitive element of the field F?. Then O/P — {0,1, A , . . . , A9"2}, the elements 1,A,A 2 form a basis in Wq over Fp. Let {/i, /2, f s } be a canonical basis in K over Qp corresponding to the basis {1, A, A2}; this means that /i,/2 and /s are arbitrary representatives of the residue classes 1,A and A2 respectively. The absolute value in K has a simple expression in terms of coefficients with respect to a canonical basis (see Lemma 2.1): if x 6 K, x = 771/1 + 7?2/2 + %/s, then ||a;|| = [max{|ryi|p,|7?2|p,|r?3|p}]3. An interpretation of the operator A associated with the form (2.41) as a coordinate representation of a pseudo-differential operator over K can be given as follows. Consider the following function 7 on the group of units U

of the field K. Let u 6 U, u = 771/1 + 772/2 + %/s- Then max \r)j\p = 1. _•? _ "

Set

fu\ - IP'"' 1 1,

if

I7?1 p-1' MP < !' MP < !» otherwise.

The operator 21 is,by definition^ pseudo-differential operator over K with the symbol a(x) = /9||a;||2a/37(ux),

x € K, p = const > 0.

(2.42)

64

Chapter 2

If {ei, 62,63} is a basis of coordinate representation then the matrix T appearing in relations between 21 and A equals M0', where M= and the matrix 0 = (fly) transforms the bases: 3

Indeed, it is sufficient to show that ),

(2.43)

M'71,%,%) =P~ 2 A f MCi,C2,C3),

(2-44)

where x = 771/1 + 772/2 + If then

where

(j=PNrij,

10 IP < 1.

In particular, if

|Ci|p = i, IC 2 | P <1 then

whence

IMCi,C2,C 3 )|p=p- 1 .

(2-45)

For the remaining values of Ci> Cz> Cs)

C2,C 3 )| P = l.

(2.46)

In order to prove this, it is sufficient to verify that |e2C2 + SaClIp = 1Supposing otherwise, we have £2C| +£sCl = 0 (mod p) (a congruence in Zp

modulo pZp), and either 2e2C2 5^ 0 (mod p), or 263^3 ^ 0 (mod p), which implies (see [18]) the nontrivial solvability of the equation £2C| +£3Cl = 0. We have arrived at a contradiction.

Fundamental solutions

65

Now the representation (2.43) is a consequence of (2.44)-(2.46). Let us expand the function j(u), u £ U, with respect to the characters of the group U. Each element u e U can be uniquely represented in the form u = /2my, where 0 < m < q - 2, ||1 - y\\ < 1. Let

Obviously, ^ is a character of the group U. We have

E ^-i)(«)1=0

(2-47)

Indeed, let u = 771/1 + 772/2+773/3. If |»?I|P = *> l^lp < 1, Mp < 1, then in this (and only this) case the class of the element u in O/P is identical to the class of the element 771 € P. Let 6 = Ap2+p+1. Then c^1 = 1, all the degrees 6*, 0 < i < p — 2, are different and together with zero constitute a subfield of O/P consisting of p elements. Such a subfield is unique (see Theorem 1.3). Hence it coincides with the field Zp/pZp considered as a subfield in O/P. Therefore for the above values of u in the representation u = f^-y we have m = li(p2 +p+l), 0 < l\ < p — 2, whence il>i(p-i)(u) — 1, so that the right-hand side of (2.47) is equal to p~a - 1, which equals the

left-hand side. Consider the remaining values of u. We have Imlm where m ^ 0 (mod (p2 + p + 1)). The function r(l) — exp I ————— \P2+P+1 is a nontrivial character of the additive group of classes of residues (mod (p2 + p + 1)). Therefore

and the right-hand side of (2.47) equals zero, as we needed to prove. It follows from (2.47) that the function 7 satisfies the local constancy condition (2.19). Let us calculate the function G(u) using the expression

66

Chapter 2

given in Proposition 2.5. In our case the characters contained in the expansion of 7 are #00 = 1 and some of the characters of the ramification degree 1, namely V;(p-i) with / > 0. Since 1 _ «-2a/3-l

we get (changing the index of summation) that - 1

+

By Theorem 1.14 we have for I ^ 0, j ^ 0

/

11=1 The last integral is equal to the sum of integrals over the sets (y = f?z : ||1 - z|| < 1},

0 < m < q - 2.

By the invariance of the Haar measure, the measure of each of them is (1 - 9"1)(9 - I)"1 = Q~l- Hence

Fundamental solutions

67

Substituting all this into (2.48) we find that for u = f^z, ||1 — z\\ < 1,

9-2

Eo E *(-

where 9-2

yf

(

1 P-

Note that 9-2

9-2

(2-49) m=0

The expression XP(P~IX)> x £ Zp, defines a non-trivial additive character n on 1ip/pLp. Since fiT is unramified, the sum in (2.49) equals 9-2

m=0

that is a sum of values of a non-trivial additive character x(— tr(-)) on all non-zero elements of a finite field O/P. Therefore 9-2

E xC-p-Va7") - -1m=0

Let us calculate E. We have, as above,

fp2+p+l, 10,

i f m = JV (mod otherwise,

68

Chapter 2

so that

m=N

0<m<<j-2 (mod (p 2 +p + l ) )

j=0

P-2

= 5>( 3=0

where (5 = Ap2+p+1 . Since the elements 0, 1, d, <52, . . . , SP~2 form a subfield in O/P, which coincides with Zp/pZp, it is clear that

I"1'

:

Thus we have proved that

p~a - 1 I)- 1 ),

(2.51)

where S is determined by (2.50). Now the hypersingular integral representation of the operator A follows from (2.37). Virtually the same calculations,together with Proposition 2.7,lead to

the formula for the fundamental solution E(x) of the operator 21. If a ^ 3/2 then E(x)=p-1\\x\\2a/3-1g(ux) and for «x =/^y, | | l - y | | < l ,

if tr(A N ) = 0 and

if tr(A N ) 7^ 0. If a = 3/2 then

,

zetf,

(2.52)

Fundamental solutions

69 iftr(A JV ) = 0

As before, a fundamental solution £ for the operator A is given by (2.38). However this time the expression for £ is not completely explicit since we cannot explicitly find the matrix T. Now we shall present another approach giving £ explicitly under an additional assumption (which can turn out to be superfluous). Suppose that F? has a self-dual normal primitive basis over Fp (see Sect. 1.2). Then the field K possesses a basis {61,62,63} of coordinate representation which is a canonical basis: the classes ei, 62, 63 of elements ej in ¥q = O/P form a basis in ¥q over ¥p. Indeed, the self-duality of the basis {Ai, X%, AS} and Theorem 1.2 imply the existence of such non-zero elements £12, Ci3i C23 G F, that Ai A2 = Cf2 - Ci2 ,

A: A3 = Cf3 - Cis ,

A 2 A 3 = C2P3 - C23 •

(2.54)

The group of units U contains such an element u0 that uqQ~l = 1, and the set M = {u0,u%,... , ujp1} is a complete systems of representatives for residue classes from O/P. Let si2>si3,S23 € M be representatives of the classes Ci2,Ci3jC23 respectively. Then s?. = crjf(sij) where &K is the Probenius automorphism.

Let ej, 6^,63 e U be arbitrary representatives of the classes AI, A 2 , A 3 . The identities (2.54) imply that

e°e? - (a* -

Sij)

= 0 (mod P) ,

1 < i < j < 3.

(2.55)

Consider the system of equations Wj ~ (s\i ~ sij ) = 0 ,

1 < i < 3 < 3,

(2.56)

with respect to e\ , 62 , 63 6 K. The determinant of the corresponding Jacobi matrix equals -2616263; for BJ = e° it is an invertible element of the ring O (since p ^ 2). It follows from Theorem 1.7 that there exists a solution {ej} of (2.56) with e-j - e° £ P, j = 1,2,3. Since s^ = (Tx(sy) we have Tr( ei e 2 ) = Tr(eie3) = Tr(e2e3) = 0.

On the other hand,e,- = Aj, j = 1, 2, 3. It follows from the construction of the basis {61,62,63} (and from the fact that K is not ramified over Qp ) that

|Tr(e|)|p = l ,

j = l,2,3.

70

Chapter 2

Let us construct the operator 21 just as it was done above but with the basis {61,62,63} instead of {/i,/2,/a}- In this case p = 1, T equals the identity matrix whence

S(xi,x2,x3) = E(xiei + 2:262 + 2:363). Repeating the above arguments we obtain for G and E the formulas identical to (2.51), (2.52), (2.53), with p = 1, and tr (Af +1 ) instead of tr (A N ). The formulas for the fundamental solution will become more transparent if we express the condition tr(AJ v+1 ) = 0 for N related to x € K, x = x\e\ + 2:262 + 2:363, Xj € Qp, in terms of the coefficients Xj. Lemma 2.2. Let ux - e^y , ||1 - y\\ < 1. The equality tr(A^ +1 ) = 0 is equivalent to the inequality , ]x3\p}.

(2-57)

Proof. If N = p or N = p2 then tr(A^ +1 ) = 0 because the basis {Ai,A 2 ,A 3 } is self-dual, A2 = A^,A 3 = xf . Consider the set 97 = kertr C F9 which is a vector subspace in ¥q over Fp. Obviously dirndl < 2. On the other hand, the vectors Af € 9? and A^ €97 are linearly independent p over Fp: otherwise it would mean that A^ ~ € Fp which contradicts the description of Fp\{0} as the set of elements Aj , i/ = 0, 1, . . . ,p— 2. Hence,the set {a: G K : tr(A x +1) = 0} consists of those x for which

Aj

= criXiX-2 +

This is equivalent to ux = §262 + 8363

(mod P)

where §2,83 E. Qp , max{|s2|p, \ss\p} = 1- It means that x = p~m(s2e2 + s3ez) + x' ,

\\x\\ = qm , \\x'\\ < q,m

(for an unramified extension (3 = p). This equality means that the coefficients of the expansion x = x\e\ 4- 0:262 + 2:363 satisfy the inequality (2.57). •

c) Quaternary forms As we know (see Sect. 1.4), a quadratic form /i(£i,£2,£3,£4) satisfying the ellipticity condition (2.40) is unique up to a linear isomorphism. It may be written as

Fundamental solutions

71

where s G Z is a quadratic non-residue mod p. Let .ftT = ( ^2 j be the quaternion algebra over Qp, with the basis 1, i, j, k and the multiplication table i2 = s , j2 = p , ij = -ji = k , k2 = -sp,

ik = —ki = js , jk = —kj = —ip. K is a central division algebra over Qp , with the normalized absolute value

||a;|| = \h(xi,X2,x3,x4)\p ,

x = xi + x2i + x3j + x±k.

In this case q — p2 , 0 = j. The reduced trace t : K -> Qp is t(x) = The basis of coordinate representation can be chosen as

It is easy to show using Lemma 1.1 that //i = — 1, //2 = 1, ^3 = — s"1, 2 M4 = s, c=l. Taking 21 = £>£/ ,

T=

/-I 1

0 1 0 0 1 0 ,

.

,

o -i o i | ' l detT lp = 0

s

0

s,

we obtain a hypersingular integral representation of the operator A with 1 — pa fl(u) = ————— T. The fundamental solution can be written as ~~ e x +, —-—— e2 0:4 -

Since there are no quadratic forms over Qp satisfying (2.40) with n > 5, the above results complete the proof of Theorem 2.3. •

72

2.8

Chapter 2

Green functions of elliptic operators

As in the previous section, we shall consider the pseudo-differential operator A with the symbol |ft(£i, . . . £n)|p% where a > 0 and ft is a quadratic form satisfying (2.40). As in Sect. 2.5, we shall call a fundamental solution £\(xi ,... ,xn) of the equation

(A + \)z = f,

A > 0,

(2.58)

a Green function of the operator A. The case n = 1 is of course contained in Proposition 2.8, and we have to study the cases when n = 2, 3 or 4. The method of coordinate representation developed above carries over virtually unchanged to the case of Green functions. In particular, the formula (2.38) remains valid with S\ substituted for £ , and E\ for E respectively. The quantities necessary in order to use Proposition 2.8 and the formula (2.38) have all been found in the course of proving Theorem 2.3. We start with the following qualitative result. As before, we assume in this section that p ^ 2.

Theorem 2.4. // the quadratic form ft satisfies (2-40), then there exists a Green function £\(xi ,..., xn) of the operator A, which is locally integrable, continuous for (xi, . . . ,xn) ^ 0, non-negative, and tends to zero at infinity. If f € 2)(Qp), then the function x = S\ * f satisfies the equation (2.58), where A is understood as a hyper- singular integral operator (2.37). The proo/follows from the above considerations. In particular, the nonnegativity of S\ is a consequence of Proposition 2.8 and the non-positivity of the function G appearing in (2.37) for each type of the quadratic form ft. The inequality G(u) < 0 is evident from explicit formulas for G found in the proof of Theorem 2.3. •

The results regarding the behaviour of £\ near the origin and infinity are collected in the following theorems. Theorem 2.5. Let n - 2, ft(£:,6) = £? - r£$. If T = e, \e\p = 1, and e is not a square, then £\(xi,X2) depends in fact on < x >e= x\ — e~lx\\p. £\ is continuous at the origin if a > 1; otherwise

a;>e In,) , < x >e-> 0. As < x >e-> oo , .3

a = l-. ,

Fundamental solutions

73

If T = p then £\ depends on < x >p= x\ — px\\p. The behaviour of £\ near the origin is as above, with < x >p substituted for < x >e. As < x >p-> oo , _

+1

i

A -2 , ^-a-1 ,(-) I ^_ 1 )A < x >p +u (< x > p

If r — ep, the asymptotic behaviour is the same as for T = p, with

< x >£p= \x\ — spx\\p substituted for < x >p. Theorem 2.6. Let n - 3,

where \£l\p = |e2|p = |e3|p = 1, e2^| + e3$ ± 0 for \&\f + \£3 p / 0. // a > 3/2, then the function £\ is continuous at the origin; otherwise 2a

3

, \0 (< x > ~ ) , ^,,I ) , n ;( n| l o g^< x > 2,x3) = < [O

0 < a < 3/2; , /0 a = 3/2,

< x >= max \Xj „ —> 0. j = l,2,3

For < a; > > 1 , Ci - 2a - 3 <£:A(a;i,a;2,a;3) <<7 2 < a r > ~ 2 a " 3 ,

Ci,C r 2 >0. (2.59)

If there exists a self-dual normal primitive basis o/Fpa ouerF p , iften

p2a+3 _ I

x >-2"-3 +0 < x as < x >—> GO, where

[-1,

Theorem 2.7. Let n = 4,

if < x > = \xi\p.

74

Chapter 2

s 6 Z is a quadratic nonresidue mod p. The Green function £\(xi,X2,x3,x^) depends on < x »= spx\ - px\ — sx\ + x\\p. £\ is

continuous at the origin if a > 2; otherwise . 2,X3,X4)

/0(«z»«-2), \ _. [O(|log |),

= <

0
-> 0. 4s < a; >-> oo , r)a+2cr)0f

_ i\

"2

"a"

Among all the assertions of Theorems 2.5-2.7 there is only one still requiring a proof, namely the lower bound (2.59) in Theorem 2.6. By virtue of (2.38) and (2.25), it is sufficient to show that the expression in square brackets in (2.25) with q = p3, v = 2a/3, TO = 1,

p~a - I Ooi = 1 + -4h——rr.

= { p2+p+l'' k O,

for the remaining / < p3 — 2,

is bounded from below by a positive constant independent of x. Thus we have to obtain a lower estimate for the expression

p~a (p_i)(-«x),

(2-60)

v-\

where ^ I /( p -i)(-u :c ) = exp the proof of Theorem 2.3), =

/

, ^V depends on 2;, 0 < 7 V < g - 2 (see

Fundamental solutions

75

We find that

"~2

"^'

(2mv(k-N). P -15—TT^ - 1

ex

v=0

The sum with respect to v is different from zero (and equal to p2 + p + 1) 9-2

only for k = N (mod (p2 +p+1)). On the other hand, £) % (p~lf$)

= -1

fc=0

as the sum of the values of a non-trivial addtitive character on the non-zero elements of a field. Thus

(2.61) j=o As we know (see (2.50)), the sum in (2.61) equals either p — 1 or —1, depending on the value of N, whence

=i
= 1.

Substituting into (2.60) we find that „ " -

2.9

Hyperbolic equations

In this section we shall consider the equation Az = /, where A is a pseudo-differential operator over Q™ with the symbol -Pa(£i, . . . ,£ n ) = i > • • • >£n)|p> corresponding to the quadratic form

M6,.. • , & ) = £ -0(6,- •.,&>),

(2-62)

5(6,..-, U 7^0 i f ( 6 , . . - , ^ ) ^ 0 .

(2.63)

As we know, quadratic forms (2.62) satisfying (2.63) exist only for n < 5. The case when n = 2 is quite simple. For example, if h(£i,£2) =£i~ £f, then just as for the wave equation over IR2, A can be transformed (by the

76

Chapter 2

obvious change of variables) into the product of two one-dimensional Daoperators. This leads to a hyper-singular integral representation

-u(x - 9,t + 6) — u(x - y,t - y) + u(x, t)] dy dd and the formulas for a fundamental solution (understood just as in the elliptic case):

if a ^ 1 ,or

if a = 1. The case when n > 2 is more difficult because a solution may be a true distribution, and the action of A on 2?'(Qp) is not defined. Denote

= {V € P(Q^) : ^(0 = 0 , £ 6 C).

This space is a closed subspace of T>(Q™ ). Of course, the Fourier transform does not preserve Dc-(Q"). However for the operator A = F~lPaF the opposite is true. Proposition 2.9. The operator A acts continuously from X>c(Qp) io Proof. Let y? € X> c (Qp)- Then ip € X>(Qp). Suppose that

C {^ 6 Q

: max |^-|p < p^},

N,leK We may assume that N > 0, I < N - 1. Let us show that there exists such v € Z depending only on AT and Z that .••,

= 0

if

Fundamental solutions

77

It is sufficient to consider only £1 , . . . , £n with max |£j p < pN . Denote l<.3
= PNtj, j = 1,... ,n. Then jj,- € Zp. If |fc(&, . . . ,£ n )| p < pv then '?i,--.,'7n)| P <^- | - 2 J v . There are two options: either max

implying that max (£,•[„ < pl and <J5(£i,... ,£„) = £>(0, . ..,0) = 0,- or l<j
at least for some j \rij\p > pl~N+l. Suppose for example that \r)i p = pk, l-N + l
max \rf> - ^ \p <

Set $=?-"!# ,j = l,...,n. Then ( # , . . . , £ ) € C7, 2N+V-1-1 P

~

Now if we choose v < 2/-4A r +l then we get v < Ik -IN (the condition for validity of the preceding construction),

whence ^i,... ,^ n ) = 0. The function Pa is locally constant on a neighbourhood of the support of $. This implies the inclusion Pa

(Q£). Hence Aip € T>c(®%)- The continuity of A follows from the definition of topology in T>c(Qp) induced from

The conjugate A' : T>'c(Qp) -> 2?c(Q?) is an extension of the operator A. Note that the Fourier transform is not denned on £>^(Q") since it does not preserve £>c(Q£)The convolution of test functions and the inversion ^(a;) = 'C(Q%) of the equation A'E = 8. For a fundamental solution E and a test function / € 2?c(Qp) the function z = E * / considered as a distribution from T>'c(Qp) satisfies the equation

A'z = f.

78

Chapter 2

In order to construct a fundamental solution E G X>^(Q") it is sufficient to consider the function

as a distribution from £>'(Q™ \ C) and to set E = FP_a,

(2.64)

where the Fourier transform acting isomorphically from X>c(Qp) to £>(Qp \ C) is extended to the action Z>'(Q£ \ C) ->• X>'(Qp) (note that the functions h and PA are even, so that here we need not distinguish between the direct and inverse transforms). According to Theorem 1.15, the function P\, X € (0, oo), with values in 2?' (Op* )> admits analytic continuation to a meromorphic function on C. Its only real poles are at the points A = — 1 and A = — n/2. This implies the following result.

Theorem 2.8. Suppose that a ^ 1 and a ^ n/2. Then the fundamental solution (2.64) can be extended to a distribution from £>'(Qp).

2.10

Cauchy problem for an equation of the Schrodinger type

2.10.1. Classes of functions. In this section we consider the initial value problem

(Az)(x,t) = f ( x , t ) ,

xe
z(x,0) =ZQ(X),

(2.65) (2.66)

where n > 1, / € E>(Q£+1), z0 £ Z>(QP),

(Az)(x, t] = F^T)_(Xtt) [\T - (a^* + ••• + a^l) \^F(v,g)^>r}z] , (2.67) a > 0 , O ^ O j 6Q P 0' = !,... ,n). As in the cases of elliptic and hyperbolic operators, our first task is to find a hypersingular integral representation of the operator A, which would extend it to a wider class of functions. In the present case two such representations will be given. One of them maintains the Fourier transform

Fundamental solutions

79

with respect to the spatial variables (thus the operator remains pseudodifferential with respect to them). The second representation resembles the ones appearing in the preceding sections, being hypersingular integral with respect to all the variables. Denote by Ba the class of such measurable complex-valued functions z(x, t) , x e £ , t € Qp , belonging to L2(ty%) for each t , that the partial Fourier transform v ( y , t ) — Fx->yz(x,t) possesses the following properties. There exist N , I € Z and a positive function c(t) (depending on u ) such that

v(y, t) = 0 for all t € Qp if \\y\\ > PN ;here \\y\\ = max \yj\p ; (2.68)

c(t)\t\-a~l dt < oo ;

\v(y,t)\ < c(t) , y £ Q£ , * 6 Qp ; j

v(y, t + t')= v(y, t) for all y 6 Q£ , t € Qp , if \t'\p < pl .

(2.69)

(2.70)

Note that the conditions (2.68) , (2.70) imply that

z(x, t)~ I xp(-x • y)v(y, i)dy,

x-y - xiyi +••• + xnyn,

Q?

is jointly locally constant: there exists such a number v 6 Z that U\ X ~\~ X

t ~\~ t \ ~~~~ U( 3* ti

for all x e Q™ , t e Qp if \\x'\\ < pv', |t'|p < j f . Lemma 2.3. If z £ Ba then the expression

••• + anyl))v(y,t- T) - v(y,t)] dr (2.71)

exists in the sense of absolute convergence of the double integral. If z £ thenA1z = Az.

80

Chapter 2 Proof. Due to the condition (2.68) the integration in the inverse Fourier

transform is conducted over the set { \\y\\ < pN}. If |T|P < p~Nl , [|y|| < pN ,

where NI = 2N+max.i <j p'1} , li = min{/, — N I } . So the integral contains no singularity,and the absolute convergence is guaranteed by the condition (2.69). 1 then

(AlZ)(x,t)

in accordance with (2.8). Observe that

After the change of variables we come to the expression (2.71). • In a sequel we shall omit the subscript identifying the operator A with its extension A\ when the latter exists. Let us introduce a more narrow class Ca of functions for which the Fourier transform can be eliminated from the definition of the operator A. The class Ca consists of functions z 6 Ba possessing the following additional properties. For every z £ Ca there exist such finite functions a(t) > 1, b(t) > 0 (t G Qp) depending only on \t\p that:

z(x,t)=Q \z(x, t)\ < b(t)

if \\x\\ > a(t) ; for all x e Q™ , t €
I [a(t)]nb(t)\t\-a-n^-ldt

< oo.

(2.72) (2.73)

(2.74)

We shall say that the integral / f ( x ) dx of a locally integrable complexvalued function / is conditionally convergent if the limit / = lim N-KX

f ( x ) dx

/ J \\X\\
N

Fundamental solutions

81

exists in C. Then by definition

j f ( x ) dx = I. Lemma 2.4. If z € Ca then

(Az)(x,t) i

a

n

~

r I " '"

I

^^

n XP

-Ir0""/2"1 dr

[*(* - y, * - r) - *(*, «)i ^ (2.75)

~

fi/ze function Ap was defined in Sect. 1.8). The integral is understood as an iterated one, with the conditionally convergent internal integral. Proof. Denote the internal integral in (2.75) by w(x,t,r). Then for r 7^ 0 we have w(x,t,r) — wi(x,t,r) - wz(x,t,T) , where W!(x,t,T) = I { n*P ( T2-} >z(x-y,t-T)dy 4 U=i V4rVJ

(the integral is absolutely convergent due to the condition (2.72)),

w2(x,t,r) = z(x,t) f[fXp

j=1J

\-]j-\ 4ra dVj

V v

We have used the formula (1.39) and the relation Xp(c2a) = Xp(a). Let us study some properties of w(x, t, T). For fixed x, t and T p > pk we have the estimates

\wi(x,t,r)\<

\z(x-y,t-r)\dy \\x-y\\
< c[a(t - r)]n b(t - T) ,

82

Chapter 2

so that max

| w(x, t, r) | <

(2.76)

where a positive constant C depends on x,t and k. If I r IP < P" then due to the local constancy of z

w x„, , t* T , r\ )__

f

y/

" JU1 TT

• -y,t-r) - z ( x , t ) ] d y

4

J

where t,T)=

/ J

w2,v(x,t,T) = z(x,t)

r TTxpMr^— " ( y1- \} dy.

I J

~ll

\4Taj/

Fix x,t and consider the function $(y,r) — z(x — y,t — r), \\y\\ > P" i T p < P" • Since $ has a compact support on its domain of definition and is locally constant we come to the conclusion that $ can be represented in the form m

•) , 1 < m < oo , t=i where cj; 6 C , Ofe(?/,T) is a characteristic function of a set

where ||j/^|| > pv , T^\P
Fundamental solutions

83

Using (1.38) we obtain that Witt,(x,t,T) = 0 if |r|p < minp2"k

It follows from (1.38) and (1.39) that W2,v(x, t, r) = 0 if r p < p2" min \a,j ~l .

Hence w(x,t,r) = 0 when T p is small.This,together with the inequality (2.76), guarantees convergence of the external integral in (2.75). Let us show that the right-hand sides of (2.71) and (2.75) are equal. Denote as before, *.(,, /\ — P1

~ / ~ f\

"\yi ^) — * £-4y^V'* / > /'

Then we obtain using (1.39) that ~*

fi

i(x,t,T)=F-^xlv(y,t-_ | T |n/2

- T

Ta £) j J

|a

^/2)J a;2 ) ) v ( y , t -

Note that for any h € Qp , ft ^ 0, Ap(ft)Ap(-ft- J ) = 1 .

This identity is an easy consequence of the well-known properties [59] of the Legendre symbol. Substitute the expression for w = wi —w2 into (2.75). Then to reach the required formula (2.71) it remains only to interchange the inverse Fourier transform and integration with respect to T. This is possible because w(x,t,r) = 0 when r stays in a neighbourhood of zero, and v ( y , t ) satisfies the condition (2.69). •

2.10.2. The inhomogeneous equation. Let us consider the equation

84

Chapter 2

(2.65) with / e £>(Q™+1). Let z(x,t) be the function

z(x,t)= *_ \__ _

//<--**->{iW£)U

(2.77)

Q

if a 7^ 1, or

i™

OS "p

\ fi "i / i

/,

(2-78)

if a - 1 . Lemma 2.5. TTie integrals (2.77),(2.78) exist as iterated ones and define a function z € Ca satisfying the equation (2.65). Proof. We confine ourselves to the case a ^ 1. The case a = 1 can be considered in a similar way. Denote by w(x,t,r) the internal integral from (2.77): x, t, r) = I f(x - y, t - r)

[ Xp

- -

dy.

(2.79)

The integral (2.79) is absolutely convergent since / € T>(Qp+1). Just as in the proof of Lemma 2.4, we can construct a finite covering of supp / by non-intersecting polydiscs obtaining the representation °k *=1

where & is the indicator of the set

Fundamental solutions

85

Calculating the integral (2.79) with the use of the identity (1.38) we easily obtain that for small |r|p, which implies convergence of the iterated integral in (2.77). This calculation also shows that

w(x, *, r) = 0 if \\x\\ > Ml + |*|p),

(2-81)

with some /j, > 0, which does not depend on r. The property (2.81) enables us to calculate v(y,t) = Fx~+yz(x,t) by interchanging the integration with respect to r and the Fourier transform. Using (1.39) we obtain that

(2.82)

where J(y,j) = Fx^yf(x,s) . Since / € P(Qp+1) , it follows from (2.82) that v € Ba with c(t) = 1\t\p~l > 7 > °> for I*!P > LThe equality (2.81) implies that z(x,t) = 0 if ||a;|| > fj,(l + \t\p) , p. > 0.

(2.83)

The properties (2.80) and (2.83) mean that z £ Ca . The representation (2.82) can be written as

• • • + anyl)),

(2.84)

where (py(t] = Xp(t(aiVi + • • • + any%))f(y,t), ^~a"> is an order a (fractional) integral of the function tp, that is ^~~a^ = fa * V, and fa is the Riesz kernel. Substituting (2.84) into (2.71) and using Proposition 2.2 we obtain that

(Az)(x,t) ••• + anyl)} f \ J

~P dr

-r

=

= f(x,t).

86

Chapter 2 2.10.3. The Cauchy problem. Of course, the formulas (2.77) and

(2.78) can be interpreted as convolutions of a fundamental solution with the right-hand side of (2.65). However, in this case we can do more than that, and now we are ready to consider the problem (2.65)-(2.66) where / £ I>(p+1) , z0 € 2?(Qp). We look for a solution z £ Ba (if possible,,? 6 Ca) understanding the operator A in the sense of (2.71) or (2.75).

Theorem 2.9. A solution z E Ca of the problem (2.65)-(2.66) is given by Z = Zi + Z2

where zi(x,t) is equal to the right-hand side of (2.77) (if a ^ 1) or (2.78) (if <* = !),

Proof. According to Lemma 2.5, z\ E Ca, Azi = f . Since zi E Ca, we have in particular zi(-,Q) E 2?(Qp) so that the function VQ(X) = ZQ(X) — zi(x,Q) also belongs to £>("). Let us investigate properties of z2. Using the identity (1.39) we find the Fourier transform w(y,t) — Fx^yz2(x,t): w(y, t) - Xp(~t(aiyl + ••• + anyl))v0(y),

(2.85)

where VQ is the Fourier transform of VQ. It follows from (2.85) that z2 G Ba with c(t) = const. On the other hand we can write

k=l

where ck 6 C , pk(x) is the indicator of the ball Vk = {y € £ : \\y-y(k)\\ < p"}, and the balls Vk do not intersect each other. Now the identity (1.38)

Fundamental solutions

87

yields

fc=l

..

n

m

re

where

, if 1(1

Hence 02 € Ca with a(t) = /z(l + |t|p) for some fj, > 0 , b(f) = C(l + (~< v, n / , o >u. Substituting (2.85) into (2.71) we obtain that Az-i = 0 . It also follows from (2.85) that z2(x, 0) = VQ(X) . It remains to use Lemma 2.5 to complete the proof. • \ \-n/2

Now we shall prove the uniqueness result.

Theorem 2.10. If z £ Ba, Az = 0, z(x,0) = 0, then z(x,t) = 0. Proof. It follows from (2.71) that (i/, *)] dr = 0 (2.86)

for all y 6 Q™ , r € Qp, where as before v(y,t) = Fx^yz(x,t). Fixing y consider the function ifiy(t) = X P (i(oi2/i + • • • + any^))v(y, t) . The equality (2.86) can be written as

By Theorem 2.1 this means that tpy does not depend on t . Since ipy(0) — v(y,0) = 0 we come to the conclusion that i/>y(t) = 0 whence v ( y , t ) = 0 and z(x,t) = 0. •

88

Chapter 2

2.11

On some models of p-adic quantum theory

The conventional description of the physical space-time is based on the use of the field E of real numbers. In most cases mathematical models based on E provide quite satisfactory descriptions of the physical reality. However one should not forget the fact that the result of a physical measurement is always a rational number, so that the use of E, a completion of Q, is not more than a mathematical idealization. On the other hand, by Ostrowski's theorem, the only reasonable alternative to E among completions of Q is Qp for some p. Other related structures, which can be seen as alternatives to the structures based on E, are adeles, non-archimedean division algebras etc., which employ Qp as their main building block. Thus it looks quite natural to try non- Archimedean models in various

physical situations, where the conventional space- time geometry is known to fail, for example in the attempts to understand the matter at sub-Planck distances or time intervals. In order to do that, the first task is the development of non- Archimedean counterparts of quantum mechanics and quantum field theory. There are two different approaches to the p-adic quantum mechanics: a wave function defined on Qp or Q™ may take its values either in Qp (or its extensions), or in C. This section is not, of course, a review of non-Archimedean quantum mechanics, which deserves at least a book, and fortunately a reader can be directed to three of them [154, 68, 69] containing many additional references, and also historical information. Our task here is much more modest: to indicate some areas of mathematical physics related to this chapter. Correspondingly, we shall deal only with complex-valued wave functions. The opposite case is expounded in [68, 69]; for some recent new approaches (which lead also to new results in non-Archimedean analysis) see [82, 86, 89]. The formalism of the p-adic quantum mechanics with complex-valued wave functions is based on the direct construction of the unitary propagator U(t) on the Hilbert space -La(Qp) related to the classical dynamics via a representation of the Heisenberg-Weyl group. Both time and spatial

variables are assumed to be p-adic. For the free one-dimensional particle one gets (with the appropriate choice of units)

=

Kt(x -

Fundamental solutions

89

where Note that the Fourier transform

Kt(z) = has the simple form Kt(z) = \p(—tz2}. The standard Hamiltonian approach fails in the p-adic situation since Qp is totally disconnected, which excludes any possibility to define infinitesimal generators of space or time translations. Nevertheless there is a natural desire to find some equation for wave functions. The first attempt in this direction was made in [153], where the equation D\tl> — D^ = 0 was considered with this purpose. Unfortunately, the solution of the Cauchy problem for that equation is not unique, even in the space 2?(Qp), which is in fact too narrow to include the wave function, whose support with respect to t is not compact. Returning to the Schrodinger-type equation (2.65) we see that the free wave function ^(x,t) = (U(t)if}o)(x) satisfies the equation Aij) — 0 with n = 1, 0.1 = 1/4, and the initial condition i^(x,0) = ipo(z), x £ Qp. Together with Theorem 2.10, this shows that (2.65) is indeed a p-adic counterpart of the Schrodinger equation for a free particle. The problem of constructing Schrodinger-type equations for more complicated dynamics like the p-adic harmonic oscillator [154] remains open. Among various approaches to the p-adic quantum field theory there is an analogue [152] of the Euclidean approach [44], which is based on the construction of a Gaussian measure fi on the space P^(Qp) °f real BruhatSchwartz distributions, with the characteristic functional

(f e z>r("). where

r r

G(x - y)
G(x) is the Green function of some pseudo-differential operator, for example the one considered in Sect. 2.8. This application shows the importance of the positivity property of the Green function, which is crucial for the construction of a Gaussian measure.

90

2.12

Chapter 2

Comments

The Riesz kernels over non-Archimedean local fields were studied in [39, 128, 143, 157] as the main examples of homogeneous distributions. The most detailed treatment (for K — Q p ) was given by Vladimirov [150, 154], who also initiated the systematic study of the fractional differential operator Da. In particular, he proved Theorem 2.1. The connection with zeroes of the Riemann zeta (Theorem 2.2) was found by Haran [52]. The results of Sections 2.5-2.9 were obtained by the author [79, 80, 87]; here we have corrected a minor error contained in the proof of Theorem 2.4 given in [79], and also some misprints in the formulas in [79, 87]. Lemma 2 appeared in [144]; our proof is taken from [79]. The first result on the asymptotics of the Green function (for n — 2, a = 1, and the quadratic form ^(^1,^2) = £1 + £2) is due to Bikulov [12]. In particular, if p = 3 (mod 4), this form is anisotropic, and Bikulov's theorem (proved by a different method) is a special case of Theorem 2.5. The asymptotics of the Green functions for some non-elliptic pseudo-differential operators over Qp was found by Jang [62]. The idea of using the spaces of functions and distributions defined outside the singularities of a symbol (as we did in the hyperbolic case) was employed in [70] for arbitrary pseudo-differential operators. However for the hyperbolic equation we managed (Theorem 2.8, see [87]) to extend the fundamental solution onto the singular set. The equation of the Schrodinger type (Sect. 2.10) was introduced and studied by the author in [77].

Chapter 3

Spectral Theory 3.1

Fractional differentiation operator

Let us consider the operator Da, a > 0, as an unbounded operator on L-z(K). As before, K is a local field, and we shall maintain the notation of Chapter 2. We shall assume that the characteristic of the residue field of K is different from 2. The domain of the operator Da is the set of those u € Lz(K), for which ll£l|aw(£) 6 L-2(K). Since Da is unitarily equivalent to the operator of multiplication by ||£||a, it is self-adjoint, its spectrum consists of the eigenvalues XN — qaN (N € Z) of infinite multiplicity, and their accumulation point A = 0. In this section we give an explicit construction of a complete orthonormal system of eigenfunctions of the operator Da . This system of functions is indexed by several parameters and consists of the following subsystems:

~ q'-N)SK(x0 - k)X(S/3l~2Nx2),

(3.1)

where N € Z, / € Z, / > 2, e = £0 + £i/3 + • • • + £i-2/31'2, £j <E 5, e0 £ P, k€S,k<£P; ^N,k,o(x) = q(N-l)/^(qN-l\\x\\)X(-kfl-Nx),

(3.2)

where N € Z, k € 5, k $ P. As before, q is the cardinality of the residue field O/P, S C O is the full set of representatives of classes from O/P, X is a rank zero additive 91

92

Chapter 3

character; if s, t € M, s >Q, a £ O, then fl, om = <

^

t = 0,

\0, « ^ 0 ;

. fl, w(s) = < V

'

0<*<1,

\0, s > l ;

,, fl, OK (o) = <

^

o€P,

\0, a £ P.

In (3.1) ZQ is the coefficient from the canonical representation (1.7) of an element x with ||a;|| = ql~N . We shall call the functions (3.1) and (3.2) the Vladimirov functions.

Lemma 3.1. The Fourier transforms of the Vladimirov functions are as follows:

(3.3)

<^)«A-(6) - *),

(3.4)

where ^o *s ^e coefficient from the canonical representation (1.7) of an element £ with ||£|| = qN , j// ts even, -,

if I is Odd,

0^ is the quadratic multiplicative character on the finite field O/P, e is the class of an element e G O in O/P; the number GK was defined in Sect.

1.3. Proof. Integrals, which emerge in the course of computing iplN k s (I > 2), after simple transformations of variables are reduced to the Gaussian integrals /s and 76 from (1.40). The proof of (3.3) consists of a numerous use of Lemma 1.4 and Lemma 1.5. The equality (3.4) is a consequence of (1.27). •

Theorem 3.1. The system of Vladimirov functions (3.1)-(3.2) is a complete orthonormal system of eigenfunctions of the operator Da in L2 (K) . Proof. It is clear from (3.3)-(3.4) that

Spectral theory

93

so that iplNtkfl, is an eigenfunction of Da with the eigenvalue \N. It follows from (1.22) that the I/2-norms of all the functions i^lN k e are equal to 1. The orthogonality of the eigenfunctions with different N or different k is the consequence of (3.3), (3.4), while the orthogonality of two eigenfunctions with the same N and different values of I is seen from (3.1), (3.2). The proof of the orthogonality of the eigenfunctions iplN k e, and il>lN k e,, , I > 2, s' ^ e", is reduced to the calculation of several integrals of the form of I5 or J6 (see (1.40)). One should take into account that \\s' — e"\\ > q2~l. In order to prove the completeness, it is sufficient to show that for each A'" the functions iJ)lN k £ form a complete system in 1/2 (5W), SN = {£ £ K '• Mil ~ QN}- It follows from the Parseval identity that the functions M O (£) = x(£a)i where either a = 0, or ||o|| > ql~N , form a total system in LZ(SN)- Therefore it is sufficient to prove that each of these functions can be expanded into a series convergent in LZ(SN) by the system of functions tf>lN

k e, which is clearly orthonormal in 1/2 (<SW)It is obvious that

If ||a|| = g1-^, a - f3N~l(a0 + ai/3 + • • • ) , £ = /3~N^0 +^(3 + • • • ) , then we obtain for £ € SN that

Suppose that ||o|| = qn~N, n > 2. Denote by (•, -)jv the inner product in L2(5jv). We have

JV

which means that

if /

n.

94

Chapter 3 The quantity of the functions iplN

k£

with fixed A r , f c = a 0 , / = n > 2 ,

n 2

is equal to (q - l)q ~ . Therefore 2

N

On the other hand, by virtue of (1.21), -

f

-

N-l

11411=9"

which implies our assertion. •

3.2

An operator of the Schrodinger type

3.2.1. General properties. Let v(x) be a locally bounded measurable real-valued function on K. Define an operator H' on L2(K) with the domain £>(H') = V(K) setting

It is clear that the operator H1 is densely defined and symmetric. Let H be its closure. The domain of the adjoint operator H* consists of those / € L2(K), for which H*f = Daf+v(x)f e LZ(K). Here Da is understood as a pseudo-differential operator (in the distribution sense). In fact, if / 6 T>(K), then Daf & Llfc(K).

Theorem 3.2. The operator H is self-adjoint.

Proof. Denote

„01(0 !\ma, i f | K I I < i ; a„2,~ fo, if||CIIi; -lll^, if||f||>i. We have Hf = F~lMaiFf + F-lMa2Ff + vf, where, as before, Maj is the operator of multiplication by the function dj. The operator F~lMaiF is bounded and does not influence self-adjointness. Next, F~iMa2F = a2 */. Let us show that the support of the distribution 02 is contained in the ring of integers O. Indeed, let

Spectral theory

95

Then 0,2, N € T^(K) and 02,^ —> a^ in T>'(K). Thus it suffices to prove that suppo^ C O for any N. We have

If 1 1 a; 1 1 > 1, there exists such £0 £ O that x^Co) 7^ 1 (since x is a rank zero character) . After the change of variables rj = £ + £0 we find that I K\W\<9N

whence O^N(X) = 0. This property of 02 means that for any ball Bxo — {x € K : \\x - XQ|| < 1} the subspace f)(x0) = {/ € L2(K) : f ( x ) — 0 for x $ SXo} is invariant with respect to the operator H2 — F~1Ma2F + Mv defined just in the same way as the operator H above. Moreover, H% commutes with the orthoprojector onto the subspace ¥)(x0), so that ^(^o) reduces the operator H2 (see [1]). The field K can be covered by a sequence of non-intersecting balls B (»> (recall that due to the ultra-metric property two balls of the same radius either coincide or do not intersect). This leads to the orthogonal decomposition XQ

where HI is reduced by each subspace f) f^o ). The convolution with 02 is a self-adjoint operator on each of them, while the restriction of Mv is bounded. Thus H2 is self-adjoint as an orthogonal sum of self-adjoint operators. • The conditions, under which the spectrum of the operator H is discrete, are given in the following theorem. Theorem 3.3. Suppose

that v is non-negative, measurable,

locally

bounded, and v(x) —> oo as \\x\\ —> oo. Then the operator H has a compact resolvent, so that its spectrum is discrete.

The proof is similar to the one for the conventional Schrodinger operators (see e.g. [124]); the main tools are the analogues of the Riesz and Rellich compactness criteria for subsets of L-2(K}. In fact those criteria (well-known for K n ) carry over virtually unchanged to our situation. For the details and generalizations see [154].

96

Chapter 3

3.2.2. The case of a radial potential. Let us consider the case of a "radial" function v(x) = V(||a;||), where V is a real-valued function defined on the sequence {ql,l £ Z\J {— oo}}. In this case we shall give a deeper analysis of spectral properties of the operator H. As before, we assume that V is locally bounded, that is the sequence {V(q1}} is bounded as / -» — oo. Denote by f)i and ft? the closed subspaces of L2(K) spanned by the Vladimirov functions if)lN k e with I ~ I and / > 2 respectively. We have

Lemma 3.2. The subspaces Sji and ft? reduce the operator H. Proof. Let P2 be the orthoprojector onto Sj2 in L2(K). It is sufficient to show [1] that for any / € ®(H) we have P2f € £>(#), P2Hf = HP2f. 00

OO

If / = E E ci,N,k,eiplNke, then P2f = J2 E ci,N,k,s^LNke1=1 N,k,e

For

1=1 N,k,e

any ip £ V(K)

K

where (-, •) is the inner product in L2(K) (recall that Da : T>(K) -> L2(K)). Since the series for / converges in L2(K) and ^y.fc.e € ^C^)) we Set tnat

/=!

so that

E <*.".*.« (^,*,£'^

oo

a

^ / = E E cwMrtW i=l AT,fc,£

where the series converges in V(K). Since V

'(K), which results in a representation of the function h — (Da + V)f € L2(K) in the form of a series convergent in T>'(K): h

^ = E E <*.".*,*[?"" + F(||a:||)^jv>fc,e(a:). /=! JV,A,e

(3.5)

Spectral theory

97

A similar representation (with / > 2) is obtained for the function (Da + V)P2f. Note that for / > 2

Let us show that Vil>\, kQ £ fii. Indeed, let y(N)(n

,n_

(m)

-

Then V^)^^ = v^N,k,o- The operator of multiplication by V^ is bounded and self-adjoint in L2(K). The subspace #2 is invariant with respect to the multiplication by yW (as well as by an arbitrary bounded function depending only on ||x||). Therefore f)\ is also invariant with respect to the multiplication by V^N\ whence Vi[>lN ko € f)i. Now it follows from (3.5) that (h, ^lNtkt£) = Cl,N,k,s[qaN + V(ql-N}},

I > 2,

so that

p h

oo

* = E E c^AvaN + niNDj^v.M1=2 N,k,e

(3-6)

Since the right-hand side of (3.6) coincides with the expression for (Da +

V)P2f, we see that P2f e S)(H), P2Hf = HP2f. • Let HJ be the part of the operator H on the subspace Sjj, j = 1, 2. It is clear that the operator H2 is self-adjoint and possesses a complete (in #2) systems of eigenvectors iplNtkt€, I > 2, corresponding to the eigenvalues qaN + V(ql~N).

Let us consider the operator H\ . Fixing v € Z denote

Denote by ftit-. and 55i,+ the closed subspaces in $ji spanned by the functions ^jy k 0 with N < -v and N > — v + 1 respectively. If JV > 1 — i/, then il>lN0(x) = 0 for ||ar|| > g", that is WvV/v,*,o = 0 if AT > 1 - j/. This means that (H1-Vv)f

= Daf,

/€5i 1 + .

(3.7)

98

Chapter 3

Lemma 3.3. The subspaces fti,- and ?)ii+ reduce the operator HI — Vv.

The proof is similar to that of Lemma 3.2. •

Let Hit- and Hit+ be the parts of the operator H\-VV in ^i,_ and fh,+ respectively. The operator Hi>+ is self-adjoint and possesses a complete system of eigenvectors ^"j^ 0, N >l — v, corresponding to the eigenvalues qaN Qf £ne muitiplicity q — I.

If / e Sji,-, then #!,_/ = Daf + Wvf. The restriction of Da to the subspace Sji,- has an eigenbasis \ ^ko\ with the eigenvalues qaN I

' ' J jv<—i/

of the multiplicity 0; — 1; therefore it is an operator of trace class. Of course the operator of multiplication by the function Wv is self-adjoint. The above considerations can be summarized in the following lemma. Lemma 3.4. The decomposition L%(K) = #!__ © #1)+ © ^2 corresponds

to the following decomposition of the operator H:

H = (VV + (Wv + A) © #!,+) © H2, where Vv is the operator on f)i = fli t _ ®fli,+ of multiplication by the function 14(11x11), Wv is the operator on #1,- of multiplication by the function Wt,(\\x\\), A is a non-negative trace class operator on #1,-, HI,_ and H? are the diagonal operators described above. Since the spectrum and the eigenfunctions of the operator H2 are known, in order to study spectral properties of the operator H it suffices to investigate the operator

Hi=Vv + (Wv + A) © Hi,+ .

(3.8)

Lemma 3.5. The essential spectrum of the operator Wv on^Ji,- coincides with the set of finite accumulation points of the sequence {V(ql)}i>v. Proof. Observe first of all that the operator of multiplication by any real-valued bounded function of ||x||, which equals a constant for ||a;|| < g", acts as a bounded operator on #!,_ (since obviously it is defined and bounded on fh,+ and #2)- This means that any point w 6 E, which does not belong to the closure of the range of the function V(||ar||), ||x|| > 5", is regular for Wv: the resolvent (Wv — w/)"1 is the operator on S)it- of multiplication by the function __, ... ... —— . Wv(\\x\\)-u Suppose that u is one of the elements of the sequence {V(ql)}i>v, that is u — V(ql°) — Wv(ql°), IQ > v. Then ui is an eigenvalue of the operator

Spectral theory

99

Wv. Indeed, since x is a rank zero character, ,i

if \\x\\ >
_ f 0,

For a fixed N the functions i/>ffko(x) (k 6 5, k g P) are linearly independent on the set {x £ K : \\x\\ = q~N+l}, since their Gram determinant (computed with the use of (1.27)) is equal to q-l -1

-1 q-l

... ...

-1 -1

-1

-1

...

q-l

- 1

(see [98]; note that the number of rows of the determinant equals q — 1). Therefore every function of the form £) cki/}^_lo k 0' where Ck € C, \ck\ 7^ 0,

Y^

c

k = 0, does not vanish identically, its support is

contained in the set {x 6 K : \\x\\ = ql°}, so that it is an eigenfunction for the operator Wv with the eigenvalue u. We can make the same conclusion regarding the value w = Wv(q") — 0. It is clear that Wv has no other eigenvalues. If a number u> is repeated in the sequence {V(ql)}i>v infinite number of times, then u> is an eigenvalue of the infinite multiplicity. Clearly all other finite limit points of this sequence also belong to the essential spectrum of the operator Wv. In order to complete the proof, it remains to show that those eigenvalues, which appear in the sequence {V(ql)}i>v a finite number of times (including possibly ui = 0) have finite multiplicities. In its turn, that would be proved if we verify that for each fixed IQ > v the subspace M(/ 0 ) = {^ € «i,_ : tp(x) = 0 for ||z|| > ql°}

is finite-dimensional. —V

Let ip € M, (p -

Y.

E

CNkipNko-

If INI = q1, I > /o, then

N--OO


-'+i,^i1-i,M(^) = 0.

N=-ook€S,k<£P

(3.9)

100

Chapter 3

Let us integrate the identity (3.9) over the set {x € K : \\x\\ = q1} using (1.27). Denoting UN = X) °Nk, we get

so that

a-w^'-1)/^-!) £ ff^W

(3.10)

JV=-00

Substitute in (3.10) / + 1 for /, multiply the resulting equality by g"1/2 and subtract it from (3.10). Then we find that

whence cr_;+i = g1/2^-;, so that for all / > IQ <7_, = <7_, 0 / 2 .

(3.11)

Suppose that all the coefficients C_; O!A (k € 5, k $ P) are given. It follows from (3.11), (3.9), and the above linear independence property, that one can recover from them in a unique way all the coefficients c-j^, I > IQ. This means that dim M < oo •. Lemma 3.5, together with the representation (3.8), enabled us to obtain a detailed information about the spectrum of the operator HI . As we know, the spectrum of HI (and H) is discrete if V(||a;||) -> oo, ||a;|| -> oo. The following result covers also the case when V(||o;||) ->• -oo.

Theorem 3.4. // the sequence {V(ql)}i>v has no finite accumulation points, then the spectrum of HI is discrete, and the operator H has a complete system of eigenfunctions. The proof follows from (3.8), Lemma 3.5, and the discretness of the spectrum of HI>+. Indeed, Wv is in this case an unbounded self-adjoint operator with a discrete spectrum, A its compact perturbation, and Vv a relatively compact perturbation of (Wv + A) © -Hi,+ . • Using (3.8) we can also obtain information about the asymptotic behavior of eigenvalues of the operator HI . Let 7V(A) be the distribution function of its eigenvalues, that is the quantity of those eigenvalues (counted with their multiplicities), which are smaller than A > 0.

Spectral theory

101

Theorem 3.5. Suppose that V(ql) ~ Cqjl, I -> oo, where C > 0, 7 > 0, £/zere exists such a natural number IQ that if hJ2>l0, h^h.

Then A ^oo.

(3.12)

Proof. Let us use the formula (3.8) with v > 10. First we shall show that the multiplicity of each eigenvalue w = Wv(ql) (I > f) of the operator Wv equals q — 1, or in other words, such is the dimension of the subspace Zi C f)i,- consisting of functions with supports in the set {x 6 K : \\x\\ = q1}. If / 6 Zi, then

\\ =ql,

where ck , c0 € C, since all the functions V'AT.A.O w^^ N ^\—l are constant for ||x|| = q1. If ||x|| = q1, then E $1-1 k o(x) = const ^ 0 as a keS,k<£P

consequence of (1.3). It means that dim Zi < q - 1. On the other hand, the set Zi contains all the functions from #1,- of the form E Ch^l_i k o> wnere E ck — 0. Next, consider the function keS,ki£P

' '

k€S,k$P

ip(x) = <5(||x|| -q1). It is orthogonal to f)z since the support of each function ify k s, j; > 2, either does not intersect, or coincides with supp
K

by Lemma 3.1. Thus ip € SJi. If N > 1 - v, then supp^)0 C {x e K : \\x\\ < q1-"} C{x£K: \\x\\ < q"}.

Since / > v, we have y> -L ^3i,+ , so that (p € f j i , - , which means that

Ck

=

102 if

Chapter 3 S

c

k = 0. Hence dim Zi = q — 1.

Let Nw(X) be the distribution function of eigenvalues of the operator Wv\ denote also

ra(A) = card{JV e Z : V(qN) <X,N> v}. It follows from the above arguments that

),

X -> oo,

where the bounded remainder term corresponds to the zero eigenvalue. By our assumption, _ 1 7

"

whence

NW(X) = -——log.A + O(l), 7

A-> oo.

(3.13)

Let {An}, {/in} be the sequences of all the eigenvalues (counted with their multiplicities) of the operators Wv and Wv + A arranged in the ascending order. It follows from the minimax principle [124] that An < Hn < Xn + \\A\\,

so that

NW(X-\\A\\}
A The eigenvalue distribution function for an orthogonal sum of two selfadjoint operators equals the sum of corresponding distribution functions. Using again the minimax principle we come to (3.12). • 3.2.3. Essential spectrum. Let us consider some classes of potentials V(||x||), for which the essential spectrum aeSs(Hi) is not empty and can be described explicitly.

Theorem 3.6.

I f V ( q l ) -» 0 as / -> oo, then aess(Hi) = {0}.

Spectral theory

103

Proof. Let Hito be the restriction of Da to the subspace fh, V,VV be the operators on 551 of multiplication by y(||a;||), Vl/(||a;||) respectively. It is sufficient to prove [124] that V(H\tQ +I)~l is a compact operator on £h. Since Vv ->• F as v -> oo, in the uniform operator topology, it suffices to prove the compactness of Vv(Hit0 + I)~l for any v. In fact we shall show that the latter operator belongs to the Hilbert-Schmidt class. The operator 14(#1,0 + I)~l has the following matrix elements with respect to the basis j iplN ko>: ZN,k-,M,i = (qaM + l)~lWN,k;M,i,

M,N £ Z; k,leS,k,l$P,

where -

J

f

1 ^ 1

, ,

~T————V M,l,0

K

It is clear that the integrand vanishes outside the set {x € K : \\x\\ < TfN>l-v,M>N,

then dx = C(qaM + 1)-^^,

C > 0, whence oo

oo

N=l-v M=N k,leS;k,l<£P 00

00

N=l-vM=N

oo

M N=l — if

M=l — v

oo \-22 ?aM + I)" < oo.

If AT > 1 - i/, M < N, then z

N,k-Mi\ ^ C(qa

+ 1)~ q

2

~

I

dx lN

M\
104

Chapter 3

so that as before, oo

N-l \ZN,k;M,l

E E E N=l-v M=-oo k,l€S;k,l

< 00.

Similarly, —V

OO Z

N,k;M,l\2

< 00.

N=-oo M=N k,l

Finally, if M < N < -v, then ZN,k;M,i
-v

1

I

dx
-v N-l ^—"^ . ,f) —,. ^ "^ ?y v ^ J k'M l\ / / \ZN ^ ^2 / Q / ' / ^ Z-^ ' * 1*1^*1*1 — * / ^ ^ £_^ V= — ooM= — oo k,l N= — oo M= — oo

E

N-l x '^

The next result deals with the situation, in which the essential spectrum of the operator H\ may have a much more complicated structure. Theorem 3.7. // V(ql) -> 0 as / -> -oo, f/ien i/ie essential spectrum of the operator H\ coincides with the set of finite limit points of the sequence (V(q1)}, I -> oo. If in addition

E IF(«')I < °°>

/=—oo

(3-14)

then the spectrum of HI (and that of H) is purely singular.

Proof. Let us use again the formula (3.8) with an arbitrary fixed number v. Let Vv,^ (n £ %) be the operator on ^i of multiplication by the function

""*"™> ~ \o,

if ||,|| < ^.

As in the proof of Theorem 3.6, we show that Vv^ is a Hilbert-Schmidt operator. Under our assumptions, VVtft -4 Vv in the uniform operator

Spectral theory

105

topology, as /j, —» — oo. Therefore the operator Vv is compact, and the first assertion of the theorem is a consequence of the formula (3.8), Lemma 3.5, and the Weyl theorem on the invariance of an essential spectrum. Let us assume (3.14). Then Vv is a trace class operator. In order to prove this, it is sufficient to verify the inequality E

I^IMDHtf-W*)!2 <**<«>•

E

(3.15)

N=-ook,leS;k,l<£Pf(

Denote by a^,k a term of the series (3.15). We have \Vv(\\x\\)\dx.

If N < -v then aN>k < CqN, so that —V

___

E E

N=-oo k

lfN>l-v, then

j \vv(\)X\ )\dx= E y r

^

r

E «W)i. whence oo

E

V^ r> > (Z/\r fc / ^* O Z—/ ' —

oo

1— N

V^ / Z—/

V^ / /—^

^V— 1 — t* /= — oc f

^ C -Z_/ / ^7 (^(ff ' x '') l——oo

/ Q /—~s N=l—v

^ ^1 Z_^ / — I——oo

and we have proved (3.15). Since Vv and A are of trace class, and HI<+ has only a discrete spectrum, it follows [123] from (3.8) that the absolutely continuous parts of the spectra of the operators HI and Wv coincide. The absolutely continuous spectrum of the operator Wv (on #i,_) coincides with the absolutely continuous

106

Chapter 3

spectrum of the operator of multiplication by Wi,(||a;||) on L%(K) (since the restrictions of the latter to f)i,+ and ftz have no absolutely continuous spectra) . On the other hand, the spectrum of the latter multiplication operator is purely singular. Indeed, its spectral projection E(&) (A C E is an arbitrary Borel set) is the operator of multiplication by the function /A(W^(|ja;||)), where /A is the indicator of the set A (see [121]). For any function / 6

\f(x)fdx,

(3.16)

*(A)

where $(A) = {x € K : Wv (\\x\\) 6 A}. In particular, let A consist of the origin and the sequence {V(ql)}i>v. Then $(A) = K. If / is an absolutely continuous vector, then -E(A)/ = 0 (since the Lebesgue measure of the set A equals zero), and by virtue of (3.16) we find that / = 0. • Example. Let F(||a;||) = sina||a;||, a € E. We have

o

o \V(ql)\ < \a\ £ ql < co,

so that the essential spectrum of the operator HI is purely singular and coincides with the set A of all the limit points of the sequence {sin at?*, / ->

00} . It is clear that sinaq1 = sin (2?r {^Q1}), where {r} is the fractional part of a number r € K. If the function 6(1) = -j^q1 is uniformly distributed mod 1 (see [93]), then the set {-j^q1} is dense in [0, 1], so that A = [-1, 1] . By the metric theorems of the uniform distribution theory [93] A = [—1,1] for almost all o € E. Some specific values of a can be found in [92, 117, 94].

3.3

Spectral properties of Da on open subsets of a local field

3.3.1. Structure of open subsets. Let G be an open subset of a local field K. In order to define the operator Da as a densely defined operator on Lz(G), we consider first the restriction of Da from T>(K) to the set T>(G) C T)(K] of test functions with supports in G. Now we are interested only in the values of (Da(p)(x) for x 6 G,

(G) is dense in L^(G). Thus Da is a non-negative symmetric operator on LI (G); denote by AG its closure.

Spectral theory

107

In this section we shall study relations between the geometry of a set G and the spectral properties of the operator AQ. We start from some results regarding the structure of open subsets of a local field. Any open subset G C K can be represented as a union N

G = (J Vk, fc=i

TV < oo,

(3.17)

of non-intersecting balls 14 = {x € K : \\x - Xk\\ < qTk}, r^ € Z. A set G of the form (3.17) is compact if and only if N < oo. The type of a compact open set G is defined as the smallest non-negative residue of the number N modulo (q — 1). It is known [136, 137, 135] that the type of G does not depend on the representation of G in the form (3.17) and determines G uniquely up to a locally linear diffeomorphism. Therefore the type is a basic geometric characteristic of compact open sets. Suppose now that G is not compact, so that N = oo in the representation (3.17). Proposition 3.1. The set G is closed if and only if the sequence {xk} has no finite limit points. Proof. Let G be closed. Suppose that there exists a subsequence {%kn}, which converges to an element £ £ K, as n —> oo. Then £ € G, that is £ € Vm for some m: Choose m in such a way that kn > m, \\Xkn — £|| <
so that Xkn € Vm, which is impossible since Vm D V*n = 0. Conversely, suppose that the sequence {xk} has no finite limit points, but the set G is not closed. The set G contains a sequence {yn} converging to an element 77 ^ G. Choose a sequence {kn} of natural numbers in such a way that all the numbers kn are different, and ykn 6 V^n . Then r^n -» — oo as n ->• oo. Indeed, if not, then there exists a subsequence {r'j} C {rkn}, such that r'j > R > -oo. For the corresponding subsequences {y'j} C {j/fcn}. {x'j} c (zfcn) we obtain

Ili4 - 411 = IK4 - <4) + 04 - <4) + 04 - 4)11 = IK - 4JI > max

108

Chapter 3

(we have used that the balls are disjoint), which contradicts the convergence of the sequence {yn}. Hence rkn -> — oo. Now

'ri">,\\vk*-v\\}^o as n —> oo, and we have come to a contradiction. •

Let us call an open set G regular if inf \\Xi — Xj\\

> 0.

Corollary 3.1. If an open set G is regular, then it is closed.

3.3.2. Model examples. Let us start the investigation of spectral properties of the operator AG from two model examples: the case when G is a ball Br = {x € K : \\x\\ < qr}, and the case of a sphere Sr — {x 6 K :

Let G = Br. It can be shown by an easy calculation that the function ib

fv~r/2> ~ JO,

X € B

^ X $ Br,

is an eigenfunction of ABT with the eigenvalue

Ao = 2+1 ^ iVa(l~r}>

(3-18)

and \\ipo\\L2(Br) = 1The Vladimirov functions iplN • e with / > 2, / — TV < r, are eigenfunctions of Da in L^(K), and their supports are contained in Br. Therefore they are eigenfunctions of AB,.. Other Vladimirov functions ^Jvj,^ ' ^ 2, vanish on Br. Similarly, the Vladimirov functions V'AT jo with 1 — TV < r, are eigenfunctions of ABr, while those with 1 — N > r are constant on Br. Hence if a function from L^(BT) is orthogonal to tpo and those Vladimirov functions, which are eigenfunctions of Asr, then its extension onto K by zero is orthogonal in L^(K] to all the Vladimirov functions. This means that ABT has a complete orthonormal system of eigenfunctions, consisting

Spectral theory

109

of •j/'Oi

with the eigenvalue AQ (multiplicity 1), w tn

^l-r,j,oiJ £ SiJ & PI

i

the eigenvalue

a 1

A! = q ( -^ (multiplicity q - 1),

Vi-rj,e (2 < / < z/,j 6 S,j £ P,e = £0 + ei/? + • • • + £;-2/?'~2) and il>l-r,j,oU €S,j <£P),

with the eigenvalue A,, = g a <"- r )

of the total multiplicity g""1^ - 1), i/ = 2, 3, . . . . In the case G = Sr we proceed in a similar way.

i

The function

1 1 2 r 2 f]
c , a; iA S r

n0,

is a normalized eigenfunction of Asr with the eigenvalue

The Vladimirov functions ipl_rj<£, I > 2, with supports in Sr, are eigenfunctions of Asr with the eigenvalue AJ = qa^~T^ . Other Vladimirov functions tplN • e , / > 2 (with l — N^r) vanish on 5V . Considering the functions ipNjfl, we see that their non-zero restrictions to Sr are proportional to the functions

1, Vj(x) - rt-JF^x),

x € Sr, j e SJ ^ P.

(3.20)

The functions (3.20) are linearly dependent on Sr since

by virtue of (1.3). On the other hand, the functions Vj(x) are linearly independent (see the proof of Lemma 3.5). Thus the dimension of the subspace ^ in L2(Sr) spanned by the functions (3.20) is equal to q — 1. Let us enumerate the elements j e 5, j £ P in an arbitrary way, so that the functions (3.20) are written as

We can choose the basis of the subspace ^ consisting of the functions 1, 1

x),

j/=l,...,g-2.

(3.21)

110

Chapter 3

The latter differences vanish for ||x|| < qr. Since they are proportional to the restrictions to Sr of the functions V'l-rjV.o ~~ V'l-j- ju t o> which vanish for ||a;|| > qr, we find that the functions (3.21) are eigenfunctions of Asr with the eigenvalue A': = <jr a ( 1 ~ r ). Thus we have found that the spectrum of Asr consists of the eigenvalue (3.19) of the multiplicity 1, the eigenvalue \( = qa^~r^ with the multiplicity q — 2, and the eigenvalues \[ = qa(l~r\ whose multiplicities can be easily computed and equal (q — l)2ql~2, / = 2,3,.... Another construction of eigenfunctions of the operator Ast will be given in Sect. 3.5. Note that in both cases (Br and Sr) the eigenvalues have finite multiplicities and no finite accumulation points. Of course, both operators are self-adjoint. 3.3.3. Invariant subspaces. Since the operator Da is non-local, the

representation (3.17) of a set G as a disjoint union of balls does not lead to an orthogonal decomposition of the operator AG- Nevertheless the above eigenbasis for Aer will be our principal tool in the general case too. It follows from the translation invariance of the operator Da that in order to obtain an eigenbasis in L^(Vk) (for a fixed fc) it is sufficient to substitute x — Xk for a;, and r^ for r in the functions ^>o> ^i-rjo-i ^l~r,j,eThe resulting functions will be denoted I/JQ(X), W-rn j f ( x ) - Note that all these functions belong to 'D(K), and their supports are contained in Vk (the function IJ)Q is assumed extended onto K by zero). This implies the density of ~D(G) in L2(G) for any open subset G C K, since the decomposition

(3.17) implies the orthogonal decomposition N

L2(G) = 0 L2(Vk), fc=i

N < oo.

Denote by f t i the subspace in L^(G) spanned by all the functions VV-rj(,,j ei I — 1> 1 — ^ — N. As we saw, these functions are eigenfunctions of Da over K; thus they are also eigenfunctions over G. Therefore Sji is an invariant subspace for the operator AG, which induces on #1 a self-adjoint operator AG,I. The spectrum of AG,I consists of the eigenvalues \ , _ f-a^-Tk)

Ai/k — <J

)

„_ I o

. I. _ 1

/u

v — 1, ^,. . • , ft — ± , . . . , j v ,

and possibly also their accumulation points. The multiplicity of an eigenvalue \vk corresponding to a fixed index k equals qv~l(q — 1). The total multiplicity should be computed taking into account that the numbers rk can coincide for certain k; thus it may be infinite if G is not compact.

Spectral theory

111

Let ftz C L2(G) be the subspace spanned by the functions ^ (k — 1, ... , N) extended by zero:

o,

x&G\Vk.

As we have seen,

(£>°^) (x) = q*+~l_ lg tt(1 - r *Vg(s).

zeF*.

(3.22)

If x e V} (j ^ A), then ^(z) = 0,

and since the balls Vk do not intersect, Ik - 2/11 = \\(x - Xj) + (xj - xk) + (xk - y)\\ = \\Xj - a;* ||,

whence

(Daipk0) (x) = 1^~_g"_1gr"/2||xj - Xfcir 0 - 1 ,

a: € J$.

(3.23)

Lemma 3.6. TTze subspaces f t i and f)2 reduce the operator AQ. Proof. It is sufficient to verify that the orthogonal projection to #1 does not send a function / £ £>(Aa) out of the domain I)(AG) of the operator

AG.

It is obvious that Da acts continuously from V(G) to Li(G); extending it to distributions we obtain a continuous action Da : L2(G) -4 V(G). It follows that AGf = Daf for all / £ £>(AG), where Da is understood in the distribution sense. On the other hand, let / € 5)(Aa). Expanding with respect to our basis we find that / = /i + /2, where N

N

and the series converge in Lt(G). Applying Da in the distribution sense

112

Chapters

we obtain the expansions, which are convergent in £>'(G):

Daf = Daf\ + Daf-2 a

rk

(3.24) l k

r,.,.:.,.f1 ^- ^h '

.

N

k=l

By the above assumption, Da f G L^G),, and it has an expansion convergent in Lz(G):

Daf = ]T ^fc^'-r^e + E^o*k,v,l,j,E

(3-25)

k

It is clear from (3.24) that

because /

n

\

/

. /

ij

\

= 0,

Now it follows from (3.25) that

so that the series for D" fi in (3.24) is a part of the expansion (3.25), it converges in Z/2(G ! ), whence (since elements of the basis belong to 2?(G))

. m Lemma 3.6 reduces the investigation of spectral properties of the operator AQ to the study of its part A(j,i on the subspace fJ2- Using (3.22) and (3.23), we can compute the matrix elements of the operator Aa,2 with respect to the basis |^Q }•:

Okk = (AG-^Q-^O) — ""^zi——T?" ~'kl; q*+i _ l

(3.26)

(3.27) 3.3.4. Self-adjointriess. Let us find some classes of open sets G, for which the operator AG is self-adjoint.

Spectral theory Theorem 3.8. adjoint.

113 If an open set G is regular, then the operator AQ is self-

Proof. By Lemma 3.6, it is sufficient to prove that AG,I is self-adjoint. Since dim^ 2 = -W, we need to consider only the case when N = oo. Let 6 be an operator on £J2 defined on finite linear combinations of basis functions by the diagonal matrix with the diagonal elements akk, T be an operator on fa defined on the same set by the matrix ( a°k • ) , where

Evidently, & — 1 C AG,I, and & is essentially self-adjoint. We shall prove that T is bounded, which will imply the self-adjointness of AG,ILet us use the Schur test [51]: the operator T corresponding to a symmetric matrix f a°^ j with non-negative elements is bounded if there exists such a sequence {hj} of positive numbers that oo

y

QjfcjlTj

\ /Z/lfc,

K — 1, 2, . . . J p, > U.

(o.2o)

Moreover, ||T|| < //. Set hj = qr'/2. By our assumption, there exists such m e Z that H^j - £fc|| > 9m f°r any 3 ^ k. It x e G\Vk, then x E Vj for some j 9^ A;, and since the balls are disjoint, m ||s_ <7 q .

We have (using (3.23) and (3.29)) oo

__

,,

a°kjhj = £ (~Da^,^) gr'/2 = - I (D^k0) (x) dx i*k G^Vk na — X1 _ * f f rt/2 ~ 1 _ -a-1 "

/" / II I 'I

n-a-lj *H

G\K fc ^

«a _ -| *

— 1_

where

rk

r I /

2

n /' _ a -l"

naa — 1

,, — * Mj _ g _a_i

(

yI

\\x-xhU>q">

\\~ _ n^

HT — -r. ll" IP ^fcll

I-"-1

^ 9Q~1 (o.^y)

114

Chapter 3

so that we have proved (3.28). • The next theorem (as well as Proposition 3.4 below) shows that AQ can be self-adjoint even for a non-closed open set G, when the centres of the balls 14 have finite accumulation points but converge to them not too rapidly.

Theorem 3.9.

If there exist such numbers a > 0, 7 £ (O, 2?°+i) ) >

\\Xi- xk\\>aq^r\

i^k;

(3.30)

, 2 a + l - 2 7 ( a + l)},

(3.31)

*=i

then the operator AG is self-adjoint. Proof. Let us keep the notations from the proof of Theorem 3.8. It is sufficient to show that the operator 1(6 + 7)"1 can be extended to a compact operator on #2 (see Corollary 2 to Theorem XIII. 14 in [124]). In fact, it follows from (3.30) and (3.31) that 1(6 + 7)"1 is a Hilbert-Schmidt operator. Indeed, by (3.26) and (3.27) its matrix elements bkj are equal to zero for j = k, while for j ' ^ k bkj = C

where C\ and C% are positive constants. It follows from (3.31) that r$ —> — oo as i —» oo. Since 77 < 1, the 00

inequality (3.31) implies the inequality ^ qTi < oo. Taking into account *=i (3.30) we find that

k,j=l

j=l

k=l

which was required. •.

Example. Let \\Xi\\ = qMi, where the sequence {M,} € Z tends strictly monotonically to -oo, and for large i

2(a +1)

q"r'
Spectral theory

115

Then the conditions of Theorem 3.9 are fulfilled - if i ^ k, then f qnM'

\\ - ) r. _ j.,11 1 i//j Jii fo j —— \

\q \

jLf

'

fc

,

i
'

> Mk > ir -^ nU -*^ nn LtU k

i > k)

for some k > 0. 3.3.5. Spectral properties. The arguments used in the proofs of Theorems 3.8 and 3.9 can be used in order to investigate the structure of the spectrum of the operator AG- Assume first that the set G is noncompact.

Proposition 3.2. Let the set G be regular: inf \\Xj -xk\\ =qm. m 6 Z.

If rk -> —oo as k —> oo, then the spectrum of the operator AG is discrete. If {p,n} is a sequence of all the eigenvalues of the operator Aa,2 arranged in the ascending order (including the multiplicities), while the sequence {rk} is arranged in the descending order, then •*^ ——————7-q y \y / —QiTn , 1 «—n—1 1 7

o n _= i1.2,... . j j

/o oo\ (3.32) \ /

Proof. It is quite clear that the operators AG,I and & have discrete spectra. Since 1 is bounded, the spectrum of AG,Z is discrete too. The inequality (3.32) is a consequence of the minimax principle and the inequality

< —9lnl_

- i_ g -a-i yf

a

i| r ||-tt-i.j ||a:| dx T -

i

1

~ v^ /" ~ i _Qg-a-i 2^ y

,/„ dx

\\*\\>1 °

~

and the proof is complete. •. Note that the spectrum of AG is discrete also under the conditions of Theorem 3.9 (the proof is actually contained in the proof of that theorem). The condition rk -> -oo is necessary for the spectrum of AG to be discrete, when G is non-compact (otherwise the set of eigenvalues of the operator AG,I has finite limit points). On the other hand, the operator AG,I always possesses a system of eigenfunctions, which is complete in Sji.

116

Chapter 3 oo

Proposition 3.3.

// ^ q~ark < oo, then the operator AG,? is compact, k=l

so that the operator AG has a complete system of eigenf unctions. Proof. The operator © is obviously compact. Since the balls are disjoint, we have \\*j -xk\\> 'v} > q(rk+r')/\ j / k, whence a,kj < Cq~a(-rk+r^/2, so that 1 is a Hilbert-Schmidt operator. • Proposition 3.4.

// -^ir-^cx,,

(3.33)

then the operator AG is self-adjoint, and its spectrum is purely singular

Proof. It follows from (3.27) and (3.33) that T is a trace class operator. Now it is evident that AG,Z is self-adjoint. The singularity of the spectrum is a consequence of the Rosenblum-Kato theorem (see [66]). • 3.3.6. Compact open sets. Let us consider the case of a compact open set G. Now in the representation (3.17) we have N < oo, and dim #2 = N. The spectrum of AG is discrete. The first natural issue for this case is the relation between geometric characteristics of the set G and the spectral properties of the operator AGIt is sufficient to consider the eigenvalues of the operator AG,\Since a ball of the radius qn in K can be presented as a union of q non-intersecting balls of the radius qn~l, we may assume that

TI = . . . = TN = r in the representation of a compact open set G in the form (3.17). Now the eigenvalues of the operator AG,I take the form qa("~T\ v = 1,2,...; the multiplicity of each eigenvalue equals N(q — l)q"~1. If q<*(v-r) < A, then v < a"1 log, A + r (A > 0). Therefore the distribution function A^i(A) of the eigenvalues of the operator AQ,I is equal to (A) =

J^

N(q- I)?"-1 = (Vol G)q.a~l l°s"

+ const,

v=l

where Vol G = Nqr = / dx is the "volume" of the set G. Thus Vol G can G be reconstructed from the asymptotic behaviour of the eigenvalues of the operator AG-

Spectral theory

117

Let NO be the type of a set G. We shall show that the type can also be reconstructed from the spectral characteristics of the operator AG . Let x be the multiplicity of an arbitrary (large enough) eigenvalue of the operator AG- Then NO is the minimal non-negative residue of the number x/(q—l) modulo (q — 1). Indeed, x = N(q — l)qk, where k is some natural number. We have -^-j- -N0 = Nqk -N0 = (N- No) + N(qk - 1) = (N- No) + N(q - l)(qk~l + qk~2 + • • • + 1) = 0

(mod (q - 1)),

and it remains to recall the definition of NO . Let A m j n be the smallest eigenvalue of the operator AGTheorem 3.10. The smallest eigenvalue of the operator AG on an open

compact set G is simple; the corresponding eigenfunction is strictly positive. The inequalities (N ^V

11 ^

g

"~1 - -

ar-M(a+l} 9

<

g-1

~

a(l-r)

9

\ .

Amln

~ 1)1fgla1_1g'-m(a+1)

(3.34)

hold, where qm = min \\Xi - Xj\\,

qM = max||a;i - Xj\\. ^r

Proof. The operator A2 is represented with respect to the basis {^0/^=1 by the matrix

where / is the unit matrix, C = (cy)

Let A m j n be the smallest eigenvalue of the operator AI . Then

118

Chapter 3

where p, is the largest eigenvalue of the matrix C. Since tr C = 0, we have H > 0, whence T . <• Q~ ,-a(l-r) ^ (fc-r-)ct A mm S ga+i < 9 > n , i——7? _ 1

1. •> 1 K ^ 1.

This means that Am;n = A m i n . Elements of the matrix C are non-negative; since all the elements of the matrix I + C are strictly positive, C is indecomposable ([57], Lemma 8.4.1). By the Perron-Frobenius theorem (see [57]) the number p, coincides with the spectral radius of the matrix C; p, is a simple eigenvalue, and the corresponding eigenvector has positive coordinates. This implies our assertions regarding Am;n. In order to prove (3.34) we shall find upper and lower bounds for \JL. Since the spectral radius depends monotonically on a matrix (see Corollary 8.1.19 in [57]), it suffices to find the spectral radius of the matrix

/O a a 0

u,

\a

I

a

» a > 0.

Its eigenvalues can be found with the aid of the formula for det(B — A/) given in [98] - they equal —a (with the multiplicity N — 1) and (N — l)a. Therefore which implies (3.34). •

Note that for the sphere G — SR we can write the representation (3.17) with N = q — I , Xj = £j/3~R (where £j, j = 1,... q — 1, are all different elements of 5, the full system of representatives of the classes from O/P, £j $. P), r = R — 1. In this case m = M = R, the inequalities (3.34) turn into equalities, and we obtain again the formula (3.19) for A m j n . The estimates (3.34) are based on the representation of the set G as a union of balls of the same radius. While such a representation of a compact open set is always possible, its use can lead to a loss of a part of the information about the impact of the geometry of G upon the quantity Am;n (for example, if for G there is a representation (3.17) with r\ 3> r^ for k ^ 1). In this connection we shall give some estimates of Am;n of a somewhat different nature. Suppose that we have a representation (3.17) with TV < oo and with numbers r^, which may in general be different. It will be assumed without loss of generality that 7"i > T-2 > . . . > 1"N-

Spectral theory

119

As before, we shall use the notation qM = max \\Xi — Xj\\. Proposition 3.5. A m j n satisfies the inequalities g"1 a(l-M) < \ . < _g ~ 1 -,«(l-n) S Amm ga+1-1 9 - qa+l _11 •

/o o r \ (*-6i>)

If n = r%, then

r1a(1-r>} -

Amin <

ir^lki -^ll—— 1 .

(3.36)

Proof. Let G',G" be compact open subsets of /if. Consider the operators ^LG/ and AQU on the Hilbert spaces L^(G') and L2(G") respectively. Let {A^} and {A^} be the sequences of their eigenvalues (including the multiplicities) arranged in the ascending order. Then the inclusion G' C G" implies the inequality A^ > A^.

Indeed, extending functions from G' and G" by zero we may assume that L2(G') C L2(G"), T>(G') C T>(G"). Since Da is essentially self-adjoint on T>(G') and T>(G"), it remains to use standard variational arguments (see Sect. XIII.15 in [124]). Now the inequalities (3.35) follow from the formula (3.18) for Am;n for the case of a ball. Let us prove (3.36). Let ^J be the orthoprojection in L2(G) onto the subspace X spanned by the functions ^Q, tpQ. Let jj, be the smallest eigenvalue of the operator AX on X defined as the restriction of the operator onto X. By the Rayleigh-Ritz scheme (see Theorem XIII.3 in [124]), in < fJ-

With respect to the basis {I^Q , IJJQ } in X

The formulas (3.22), (3.23) give that

l-a-1

so that

Ax =

q

~1 ga(1-r^I-

qa

~l

qri\\xi -ay"0"1 f°

and p, is equal to the right-hand side of (3.36). •

07'

120

3.4

Chapter 3

An analogue of the Hamiltonian of point interactions

3.4.1. Point interactions. Point interaction Hamiltonians in quantum mechanics (see [2]) are the operator realizations in L2(K") of differential expressions of the form —A + Sy, where 6y is the delta-function concentrated on the subset Y C Rn. Such realizations are usually taken to be self-adjoint extensions of the operator HO obtained by restricting the selfadjoint operator —A (with the Sobolev space H2(M.n) as its domain) to the set of functions vanishing on a neighbourhood of the set Y. More general operators, for example (—A) fc + 5y, have also been considered. In the latter case, if Y is a finite set consisting of m points, then the operator HO is essentially self-adjoint if and only if n > 4k. If n < 4k, then the deficiency index of the operator HO depends on k, and for n — 3, for example, equals (m,m) when k — 1, and (10m, 10m) when k — 2. Below we consider similar constructions for the operator Da. In order to describe self-adjoint extensions, we use the techniques proposed in [73]. In contrast to the real case, here the deficiency index does not depend on a. 3.4.2. Deficiency index. Let fi C K be an open set, such that the operator AQ is self-adjoint. Let us write a representation (3.17) for the set 0:

ft = |J V*.

N < oo,

V>c = {xeK: ||ar-a: x i|< g r -}. We shall consider an open subset

G — ft \ {xi,... ,xm}, where m < N is a fixed number, 1 < m < oo. Theorem 3.11. If a < 1/2, then the operator AG is self-adjoint, so that AG = A&. Ifa> 1/2, then the deficiency index of the operator AG is equal to (m,m). It is convenient to single out the basic technical part of the proof in the form of a lemma. Lemma 3.7. Let 5k £ £>'(fi) be the delta-function concentrated at the point Xk- If a > 1/2, then the equation (3.37)

Spectral theory

121

where Da : L^(ft) —>• Z>'(fi) is understood in the distribution sense, has a unique solution. If a < 1/2, then the equation

(3.38)

*=i 771

where p^ € C, ^ \pu\ ^ 0, has no generalized solutions belonging to 1/2(0). fe=i Proof. Let <£ e £>(fl). Expand

oo

if £

N

E^OCW-

(3-39)

Since the support supp y> is compact, it is covered by a finite number of balls Fxr, so that the series with respect to x in (3.39) are finite sums even when N — oo. Let us choose a natural number n in such a way that na > 1/2. Obviously,

where Dna(p E L2(fl), so that Z-^i

It is clear from the formulas for the basis functions that <

Now it is easy to see that the series in (3.39) is uniformly convergent: the absolute value of the general term of the series with respect to v does not exceed V

£ £

'=1 j€S,j£P

122

Chapter 3

where (since the number of terms in the sum over e equals (q — l)ql~2)

E E In particular, the equality (3.39) holds for x = Xk- Note that

(

0, g (,-r fc -i)/ 2)

& = x,/>l, fc = x ,i = i,

0,

k ± x,

k:

[0,

Therefore

- pfc/2 .

(3-40)

The equality (3.40) means that 9(l'-p'-1)/ViL*PtJ,o + 9-p'/a^,

£

(3-41)

where the series converges in P'(fi). Suppose that a function u €. L? (fl) is represented as a convergent series in TV

oo

v

_______

./V

E E^^-^^ + E^^.

(3.42)

Applying the operator D" termwise, we get a series converging in £>'(fi). Note that

(see (3.26), (3.27)). If> € 2?(ft), then

(

JV

\

N

N

E °*/« «' N

/ N

do

\

= E I E « «M ) (^o > v) ' u=l \x=l

/

Spectral theory

123

where the order of summation can be changed because the series with respect to p. contains only a finite number of nonzero terms (due to the compactness of supp<^), while the series with respect to x is absolutely convergent: {dk} € li and {a^^} € li for each fixed //. Thus

For the distribution Dau + u we get the expansion

N +

/ N

\ d a

d

/L ( Y, * »» + M PO > (3-43)

which converges in £>'(fi). Let us consider the case when a > 1/2. In order to construct a solution of the equation (3.37), we take a function (3.42), where civile = 0 for xr ^ k, and also for x = k, I > 1,

(3.44) and the coefficients d* are determined from the system of equations

(3.45)

The system (3.45) is solvable in / 2 because the operator AQ^ is selfadjoint and non-negative. Next, 00

___

OO

X! i^

so that u € £3(0)- Comparing (3.41), (3.43), (3.44), (3.45), we see that u is & solution of (3.37). The uniqueness follows from the fact that AQ is self-adjoint and non-negative. Suppose now that a < 1/2. Assume that a function u € L 2 (fi) of the form (3.42) satisfies the equation (3.38). Arguing as above, we find that --

1)/2

p

a

-1

[<7<"- »> + 1]

K > m, , x<m,l = l, x < m,l > 1,

124

Chapter 3

and at the same time £) \c^vijs\2 < oo. However for each x

whence 2

= oo

if px ^ 0, and we have arrived at a contradiction. •

Proof of Theorem 3.11. Let 0? be the orthogonal complement in Li(G) of the range of the operator AG + I. It suffices to verify that ,. ~ /O, a < l / 2 , dim 91 = < m, a > 1/2. If w € 91, then for any tp £

(u,Daip +
Then Dau + u = ^ Wfc on fl, where ujfc = (pk(Dau + u). It is clear that *=i v i , . . . , um € T>'(K) are concentrated at the points {xi,... ,xm} respectively. By Theorem 1.9

Vk=pk5k,

Pk 6 C (k = 1,... ,m).

Thus the function u € 91 satisfies the equation (3.38). If a < 1/2, then u = 0 by Lemma 3.7, and this is what was required. Let a > 1/2. Denote by Uk the solution of the equation (3.37). Obviously, Uk € 91 (k = m

1, ... , m), and the functions Uk are linearly independent. If w = ^ PkUk, k=l

thenDa(u — w) + (u — w) = 0 on fi, u — w 6 £2(0), so that u = w because v4o is self-adjoint and non-negative. Hence {ui, . . . ,u m ) is a basis in 91. • The self-adjointness of the operator AG for a < 1/2 can be deduced also from the next proposition, which contains a simple method of approximating a function from Z>(fl) by functions from V(G) (see Chapter X, Problem 9 in [122] for a similar result for the Laplace operator on En \ {0}).

Spectral theory

125

Denote , x JO, r < l , »?(»•) = < 1, r > l ,

Proposition 3.6. If a < 1/2, then for any function f 6 £>(Q) (fnf

Da((fnf)

-> /,

->• Daf,

H ->• 00,

TO the space L?(£l).

Proof. The first limit property is verified in a quite simple way: if n is large enough, then

= Y" (\

rrf j

(x)f( )-

)! 2 d

|/(o;)| cJa; —>• 0,

Let us consider the expression for Da(ipnf).

n —> oo.

We have

(Da(
-

(3 46)

'

As before, the first summand in the right-hand side of (3.46) tends to Da f in Li(£i) as n -> oo. It follows from the identity

01 . . . am - bi . . . bm - (a! - &i)o 2 ... a m + 61(02 - 62)03 . . . am + 6162(03 - 63) . . . am + 616363 . . . (am - 6 m )

that the square of the 1/2 (O)-norm of the second summand in the right-hand

Chapter 3

126

side of (3.46) does not exceed

C

j=i;

/

- 2 \ri(qn\\x

Htfir 1

K

< Cm/ K

^

^

(\n(qn\\x-y\\)-rj(q-\\x\\)\ y X

Il2/ll a+1

2

^ -I 2

/•v.__ n (l_2a) /"

ucr+i

/ I'/CIIC ~ Cll) ~ ^ ( I I ^ I D I

I y

JA

where we have still to show that the iterated integral on the right is finite. Denote

It is clear that is non-zero only for

= const if ||f|| < 1. Let ||f || > 1. Then the integrand || > 1, so that

IICIIM

Let us consider the canonical representation of an element T, r = a0 + ai(3 + a2/32 + • • • ;

a0, GI , . . . € 5, a0 £ P.

It is convenient to assume that 0, 1 € S. If ao ^ 1, then ||1 — r|| = 1, and the integrand equals zero. If ao = 1, 01 ^ 0, then ||1 — r|| = q~1,

Further, if o0 = 1, ai = ... = o/_i = 0, a; ^ 0, then ||1 — r|| = g~', that is

0,

if

Spectral theory

127

It follows from the invariance of the Haar measure that the measure of this set of elements T equals (1 — q~l)q~l .

Therefore if ||£|| =qk,k>0, then

l=k

Now we find from the above estimates that the L% (O)-norm of the second summand in the right-hand side of (3.46) tends to zero as n ->• oo. • 3.4.3. Self-adjoint extensions. Let a > 1/2. For this case we shall describe all self-adjoint extensions of the operator AQ. Let, as before, Uk € Z/2(fl) be the solution of the equation (3.37), k=l,2,...,m. Denote by #0 the subspace of 1/2(0) spanned by the functions ui, . . . , um. It was shown in the course of proving Theorem 3.11 that #0 = ker (A*G + I). For

the domain of the operator A*G (or AG + 1, which is the same) we obtain the equality

If u € £> (A*G), then u = f + T, f € 2>(4 n ), T e fto,

AGu = (AG + I)u-u = ( A f i + I ) f - u. Let *p be the orthoprojection on #0 in the space £2(0). Proposition 3.7. The domain of every self-adjoint

extension of the oper-

ator AG is a part of 1) (A*G) specified by the condition

(k-I0)y(An+I)f

+ i(k + I0)r = 0,

f + T£3(A*G),

(3.47)

where k is an arbitrary unitary operator on f)o, IQ and I are the unit operators fto and I/2(O) respectively. The proof is a consequence of general results from the extension theory of symmetric operators (see [45, 46, 96]) applied to the positive definite symmetric operator AQ + 1. • As in [73], it is easy to describe some other classes of extensions maximal dissipative extensions, solvable extensions etc.

Example. Let 0 — K, m = 1, x\ =0. For this case the condition (3.47) can be written in a simpler form. The subspace ^0 is now one-dimensional; if uo = ||ui||^1/Q|Ui, then the orthoprojection ^P has the form

tyu = (U,UQ)UO,

u

128

Chapter 3

Thus for any / £ V(K) <$(Da + /)/ = /(0)«o,

(3.48)

where we write Da instead of ^4^. By arguing as in the proof of Lemma 3.7 it is not hard to see that the expansion of any function / € 3)(Da) with respect to the eigenbasis is uniformly convergent. Hence / is continuous, and its value at a point makes sense. Moreover, the same arguments prove the continuity of the imbedding of 1)(Da) (with the graph norm) into the space of bounded continuous functions. Since Da coincides with the closure of its restriction to T>(K), the equality (3.48) holds for any / £ £>(Da). An element r £ fto has a unique representation in the form r — auo, a £ C. Denoting we come to the following description of all self-adjoint extensions. One of them (corresponding to k = 1) is the operator Da. The others are parametrized by the number c £ K. For a fixed c the domain of the corresponding extension consists of the functions u = f + auo (f £ '£(Da), a € C), for which /(O) + ca - 0. Note that the function UQ can be computed explicitly by applying the Fourier transform to (3.37) and using integration formulas of Chapter 1.

3.5

Multiplicative fractional differentiation

Let us consider the operator Da on the sphere 5i, that is on the group of units U of the field K. As we saw, the spectrum of the operator AJJ consists of the eigenvalues

qot-i _ 20-i _|_ i AO = —-——-a-i—— AI = qa ka

Xk = q

(multiplicity 1),

(multiplicity q — 2),

(multiplicity (q - 1)V~2), A = 2,3,... .

One can notice that the multiplicity of A^ coincides with the number of (multiplicative) characters of the group U with the ramification degree k (see Sect. 1.3). In this section we show that the coincidence is not accidental - the multiplicative characters are actually the eigenfunctions of the operator AU, it admits a hyper-singular integral representation based on the multiplicative structure.

Spectral theory

129

Theorem 3.12. (i) // 6 is a character of U with the ramification degree k, then AjjO = Afc#. (ii) If&T>(U), then

(Au(p)(x)

II 1 (3.49)

Proof, (i) It is sufficient to consider the case k > 1, since the case k = 0 has been dealt with above. According to Theorem 1.14, the Fourier transform

if ||£|| ^ qk . Now it follows from the definition of Da as a pseudo-differential operator that (ii) In order to prove (3.49) denote the expression in the right-hand side by Ma. It suffices to show that Ma6 = \ke.

(3.50)

for each character 8 with the ramification degree k. The equality (3.50) is obvious for k = 0. If k > 1, then

(Ma9}(x) = 6(x) \ !*"£_! / 111 - yir^lW) ~ 1] dy + A0 1 , x € U. I u ) (3.51) Consider the function

f(z) = It is an entire function due to the local constancy of a character. As in the proof of Proposition 3.6, we can consider the canonical representation of an element y € U, y = a0 + ai/3 + a2/?2 + • • • ; a0, ai , . . . € S, a0 £ P,

130

Chapter 3

assuming that 0,1 S S. It follows from the invariance of the Haar measure that the set of such y €. U that OQ ^ 1 has the measure 1 — -. On this set || 1 - 2/|| = 1. Further, for any / > 1, the set of such elements y that a0 = 1, dj = 0 for 0 < j < I, a/ ^ 0 (on this set ||1 - y\\ = q~l) has the measure (1 - q~l)q~l. Therefore if z > 0, then

In order to calculate the integral containing 8(y) we use the decomposition 00

U=\J(Vj\Vj+1), 3=0

where V0 - U, Vj = {y 6 U : \\l-y\\ < q~j], j >l- Using the identities (1.30) we find that

u

if z > 0. Thus

Due to the analyticity of / this equality remains valid for z = —a. Substituting it into (3.51) we come to (3.50). •

3.6

Comments

The Vladimirov functions were introduced for K = Qp by Vladimirov (see [154]), and for a general local field K by the author [81]. Our assumption that the characteristic of the residue field is different from 2 excludes the case K = Q2, which was also considered in [154]. Theorem 3.2 is contained essentially in the paper [60] by Ismagilov, Theorem 3.3 is a special case of results by Vladimirov [151, 154]. Other results of Sect. 3.2 are taken from [75]. A detailed analysis of the spectrum and eigenfunctions of the Schrodinger type operator Da + V(\x\p) over Qp,

Spectral theory

131

with V(r) —> oo as r ~> oo, is given in [154]; the case of a "harmonic oscillator" (a = 2, V(r) = r 2 ) was considered also by Blair [16].

The eigenvalues and eigenfunctions of Da for a ball and a sphere were found by Vladimirov [151, 154]; the general theorems of Sect. 3.3 appeared for the first time in [78] (for K = Qp), as well as the results regarding a p-adic analogue of point interactions. Theorem 3.12 is taken from [85].

Chapter 4

Parabolic Equations 4.1

An analogue of the heat equation

4.1.1. Fundamental solution. Let us consider the Cauchy problem

^^-+a(Dau)(x,t)

=f(x,t),

u(x,Q) = (p(x),

x€K,Q
x 6 K,

(4.1)

(4.2)

where a > 0, a > 0, K is a local field, the right-hand side / and the initial function

u(x,t) =

r

J

t

Z(x - f , t ) ( f > ( £ ) d £ +

f

J

dr

f

J

Z(x-£,t- r ) f ( £ , T ) d r ,

(4.3)

provided (p and / satisfy some natural regularity conditions. We set Z(x,

t) = j

x (-^) e -<"H€H

a

df.

(4.4)

K

As in the preceding chapters, we follow our notations regarding local fields; in particular, || • || is the normalized absolute value on K, \ is a rank zero

133

134

Chapter 4

additive character. As before, we shall denote various positive constants

by the same letter C. Using the integration formula (1.28) we find that

v=Q

If we expand the exponential function in a power series, change the order of summation, and sum the geometric progression, we find that for x ^ 0 00

f_1") m

1 _ f,am

m Z(x,t) =V -—'— • ———-——-(at) \\x\ -«"»-!. V ' ^ ml 1 - «-am-l V I I I I

(4.5) V I

m=l

It follows from (4.4) that Z(x, t) is continuous with respect to x € K. As we saw in Sect. 2.1, the function £ i-+ exp(—oi[|^||a) is positive definite. Thus Z(x, t) > 0 for all x, t. This can be verified also by a direct calculation - according to (4.4), 00

Z(x,t)=

.

a

Yl exp(-aV )

J

-

where ft/t is the indicator of the ball {x € K : \\x\\ < qk}. It follows even that Z(x, t) > 0 for all x, t. Lemma 4.1.

Z(x,t) < Ct(tl'a + II^H)-"- 1 ,

x€K,t>0.

(4.6)

Proof. By (4.4)

Z(x,t)<Jexp(-<*t\\t\\a)dt.

(4.7)

Parabolic equations

135

Let an integer k be such that qk~l < t1/01 < qk. Then

Z(x,t) < yexp(-aga(*-1)||C||a)de K

= q-^-V f exp(-a||»7||a) drj = Ct~l'a. K

On the other hand, it follows from (4.5) that if ||z| > i1/", then

z(X,t) < \\x\\-1 53 _(*||a;||-r < c-iHsir1. oo

,~,m

TO=1

The last two inequalities imply (4.6). Indeed, if ||a;|| > tl/a, then

If ||x|| < tl/a, then

so that t~l/a < 2a+1t (\\x\\ + tl/a]

-a-l

The inequality (4.6) shows in particular that the function Z belongs (with respect to the variable a;) to Li(K), so that we find from (4.4) and the Fourier inversion formula that

I Z(x,t)dx = l.

(4.8)

K

If ip is a locally constant function on K, and \ f ( x ) , t ->• 0. K Indeed, by (4.8)

Z(x - C, 0^(0 df = _ Z(x - £, A: and it remains to use the inequality (4.6) and the local constancy of (p.

136

Chapter 4

dZ Let us consider the derivative — . It is clear that one may differentiate t/t in (4.4) under the integral sign: (4.9)

~ K

If x ^ 0, then in accordance with (4.5)

x. Therefore we can define (by (2.8)) the function

In order to compute this function, denote

Nl<«m

Z(m) is bounded and locally constant with the exponent m in the variable x. We have

f

J

By Proposition 2.3 the inner integral equals ||/?||7, so that

According to (1.28), Z(m) depends actually on ||a;||. Fixing x ^ 0 we find

Parabolic equations

137

that

as m -> oo, by the dominated convergence theorem. Now it follows that { ff

•/• j -—-

/

I tJU j t/ t —— f

"v/j — : 7 * T ) t t l T ) t l

/V V

//I

/I

(^"VTll — / 7 / l l T ) l l r \

11/11

i flT)

/

/?

1* ~^~- 0 /"

(4

V

ifjl

/

/<-

The next lemma is proved in the same way as Lemma 4.1. Lemma 4.2.

'dZ(x,t)

< c(ti/a + \\x\\ra~i + Nl)-^1.

(4.ii) (4.12)

It follows from (4.12) and the Fourier inversion formula that

(D~xfZ)(x,t)dx i

/

= Q.-

(4-13)

K

4.1.2. Heat potentials and Cauchy problem. Denote by 9J17 (7 > 0) the class of complex-valued locally constant functions ip(x) on K, such that

If a function tp depends also on a parameter i, we shall say that if) € 9J17 uniformly with respect to t, if its constant C and its exponent of local constancy do not depend on t. Let us study the "heat potential"

t u(x, t, T) = j d6 I Z(x -y,t- 0)f(y, 9) dy. T

K

We shall assume that / € 9Jt,\ (0 < A < a) uniformly with respect to 6, and / is continuous in (y,0).

138

Chapter 4

Let us prove that u € 371 A uniformly with respect to t, T. Writing u(x,t,r) as t

U(x, t, T)

= j de j z(t, t - 9)}(x - & e) d$, T

K

we see that u(x, t, T) has the same exponent of local constancy N as /. Furthermore, by Lemma 4.1 we have t

u(x, t,r)\
K

(4.14)

Lemma 4.3. IfQ
(b + \\x-t\\ra-1\\t\\Xdt
xeK,

(4.15)

K

where the constant C does not depend on b, x. Proof. Denote by I(b,x) the expression in the left-hand side of (4.15). Let an integer m be such that qm~l
K

Let J3m~lx = y, \\y\\ = qk. We have

where

/

k-l

i(2/)= E / =

-

Parabolic equations

139

Let us find a bound for each summand. Firstly, k-l

Next,

W=i Represent an element 77 in the canonical form:

77 = o0 + ai/3 + a.2/32 + ••• , where aj £ S, ao ^ P, and the series converges in K. As in the similar calculations performed in Chapters 2,3, it is convenient to assume that 0,1 e S. Ifa 0 ^l,then||l-?7|| = 1. If 00 = 1,0! £ 0, then ||1-7?|| = a"1. If o0 = 1, ai = 0, o2 ^ 0, then ||1 — 77(1 = q~2 etc. The measure of the set of 77 € U with the above properties equals respectively l-2a -1 , (q— l)g~ 2 , (q — l)g~ 3 etc. Therefore

j=0

< \\y\\x-a j(\\y\\~1 + Ml)-"-1 dr, = c\\y\\ K

140

Chapter 4

Finally, .

/ (i

K

As a result we find that 1(1, y) < C(l + \\y\\x), whence

Now from (4.14) and (4.15) we get the required estimate of u(x,t,T):

\u(x,t,r)\
(4.16)

where C does not depend on t,r(< T), that is u € 9Jl\ uniformly with respect to t, T. As in the classical theory of parabolic equations [97], it is easy to comdu(x, t, T} pute the derivative —^——-. Let t-h

uh(x,t,T) = j d6 IZ(x -£,t- 8)f(£,6)d£, r

(4.17)

K

where h is a small positive number. We can differentiate Uh under the sign of integral, so that t-h

duh

at

r

K

r

K

K

t-h

K

K

K

Parabolic equations

141

The first integral contains no singularity at t = 6 due to (4.11) and the local constancy of /. The second integral is equal to zero by virtue of (4.8). The third integral can be written as the sum of the integrals over {£ £ K : \\x — £|| < qN} and the complement of this set; one of them is estimated on the basis of the uniform continuity of /, while the other contains no singularity. Finally, the fourth integral tends to f(x, t) as h —> 0. As a result we obtain the formula

T

K

(4.18) Let us consider the action of the operator D7, 0 < 7 < a, upon the heat potential u(x, t, T). Note first of all (see (4.16)) that the function D~ A. If 7 < a, then

Z-, = D^Z. (4.19) K

Indeed, let

and Uh(x,t,T) be given by (4.17), 0 < h < t — T. By the Fubini theorem

t-h

= j de jZ^k(x-y,t-6)f(y,e)dy T

On the other hand,

K

K

(4.20)

142

Chapter 4

where

It is clear that

inwi < jf^jj / wo - ii • iicir-1 rfc =CIMP, K

whence

h + r, k. Writing the right-hand side of (4.20) as t-h

dO

j

ZJtk(x-y,t-6)f(y,e)dy

ll*-y||>
+ j dO

j

Z^k(x-y,t-0)f(y,e)dy

\\x-y\\
(I is a fixed natural number), it is easy to pass to the limit in (4.20) for k —> oo, since for \\x\\ > q~l , k > I,

We find that t-h

(D->Uh)(x, t, T) = j d8 j Z^x - y, t - 6)f(y, 6) dy. T

(4.21)

K

It follows from Lemma 4.3 that M/J € 9Jl\, uniformly with respect to h. Together with Lemma 4.2 and the dominated convergence theorem that makes it possible to pass to the limit in (4.21) for h -> 0, which gives the required equality (4.19).

Parabolic equations

143

Suppose now that 7 = a. The equality (4.21) is valid in this case too. Using (4.13) we may rewrite it as t-h a

(D uh)(x, t, T) = I d61 Za(x - y, t - 9)(f(y, 6) - f ( x , 6)} dy. T

(4.22)

K

Since / is locally constant, we may as before pass to the limit in (4.22) for h —t 0, so that 4

a

(D u)(x,t,T) = jde j Za(x-y,t-d)[f(y,0)-f(x,d)}dy. T

(4.23)

K

Now we are ready to construct a solution of the Cauchy problem (4.1),

(4.2). We assume that

4.2

Probabilistic interpretation

It follows from (4.4) that Z(x,t1+t2)= f Z(x-y,t1)Z(y,t2)dy,

ti,t2 > 0, x e K.

(4.24)

K

Theorem 4.2. The fundamental solution Z(x,t) is a transition density of a time- and space-homogeneous non-exploding right-continuous strict Markov process without second kind discontinuities.

144

Chapter 4

Proof. The identity (4.24) is equivalent to the semigroup property for the family of operators

(®(t)f)(x)=

Z(x-y,t)f(y)dy. K

We know that Z(x,t) > 0, and that Q(t) preserves the function f ( x ) = 1 (see (4.8)). Thus Q(t) is a Markov semigroup. The required properties of the corresponding Markov process follow from the inequality (4.6) and general theorems of the theory of Markov processes [26]. • The process £a(i) constructed in Theorem 4.2 is the non- Archimedean counterpart of the symmetric stable process on M. Below when speaking about £a(t), we shall assume for simplicity that a = 1. It is often useful to consider £a(£) as a stochastic process with independent increments (see [56, 142, 49] for the basics on stochastic processes on locally compact groups). It follows from Proposition 2.3 that the Levy measure of the process £a (i) is

Now, considering Da as a self-adjoint operator on L^(K) (as in Chapter

3) we can obtain an analogue of the Feynman-Kac formula.

Theorem 4.3. Suppose that V : K —» K is continuous and bounded from below. Then for any f €

D(0,t;K)

where ^x is the measure on the Skorokhod space of paths [0, t] -> K, right continuous with left limits, corresponding to the Markov process £a starting at a point x 6 K. Proof. Let us consider first the case where V is compactly supported.

145

Parabolic equations

By the product formula for semigroups of operators

I ... I Z [x-xn,-] ...Z(x- xn,-]Z n J J \ ) V

K

i,-] f ( x i ) n

K

n-1

x exp

... dxn j=0

= lim

n—>oo

Since the paths in D(0, t; K) are right continuous with left limits, and V is uniformly continuous, the composition V ° w is Riemann integrable, and t

lg, H

->• oo.

j=0

Next, since V(x) > V0, n-l

I = e~tv°

-t

(x) < oo

D(0,t;K)

for almost all x. By the dominated convergence theorem we come to (4.25). For general V we can consider the cutoffs

Vk(x) =

V(x), 0,

if INI < qk,

otherwise,

apply (4.25) with Vk substituted instead of V (note that 14 is continuous!), and use the monotone convergence theorem for integrals, and the strong resolvent convergence of D01 + Vk (see Theorem VIII.20 of [121]) as k -> oo.

•

146

Chapter 4

4.3

Stabilization

Let us consider the Cauchy problem (4.1)-(4.2) with f ( x , t ) = 0 and

u(x, t) =

Z(x - £, t)
x&K,0
(4.26)

K

to stabilize as t -> oo. It is clear that u(x,t) belongs to 9Jt0 uniformly with respect to t € [0,oo). In particular, the solution is bounded on K x [0,oo). First we shall find another representation of u(x,t). Denote

V(x + y ) d y ,

v = 0,±1,±2, . . . .

(4.27)

Lemma 4.4. oo

u(x,t)=

^ (M^Vjlc^W-^-iWl,

x£K,0
(4.28)

L>= — OO

where

Proof. Let us rewrite (4.26) using (4.5) as oo

u(x,i) = ]>3 Qv(t)hv(x), where 00

/_~\\ n

i _ fian

n=l

hv(x) =

I


\\y\\=i" We apply the Abel transform M

5] gv(t)hv(x)= v=-N

M-l

X) ^(^^W v=-N

(4.29)

147

Parabolic equations

where M, N are natural numbers, Hf, '(x) = ^ hn(x). n=-N

Let

HIv( v x) = 5^ hn(x) "=~°°

=

I \\y\\
We find that H^x^^Cq",

\H^(x)-Hv(x)\
I

dy
\\y\\
n=l

= q-v[cv(t) - cv^

If v < 0, then

|^(i) - flv+1 | < Cq-"exp(-atq-av),

\hv(x)\ < Cqv .

It follows from these estimates that

-i

as N —>• oo. Passing to the limit TV —> oo in (4.29) we obtain the equality M

M-l v(t)hv(x)=

53 Hv(x)q-v[cv(t)-cv-i (4.30)

For v > 0 we have ( a+1 ),

| fll/ (t) - gv+1(t)\ <

148

Chapter 4

Note that q~vHv(x) = MX f. Passing to the limit M -> oo in (4.30), we come to (4.28). • Now we formulate the stabilization conditions. As before, we assume that if € 9Ko-

Theorem 4.4. In order that u(x, t) -> / (=const) as t -> oo for all x £ K, it is necessary and sufficient that M^'ip -> / as n -> oo, for every x £ K. Proof. The sufficiency follows from the representation (4.28). Indeed, the series over negative values of v on the right in (4.28) converges uniformly with respect to t £ [to,oo), to > 0, and hence its sum tends to zero as t -> oo. Let MI (x, t) = ;/=0

Obviously, N

N-l

J" V) M*) - <:„_! v=0

The series j/=0

converges uniformly with respect to i, by the Abel convergence test, since Cv(t) t 1 as v -> oo, \cv(t)\ < 1. Taking the limit in (4.31) as N -> oo, we see that ui(x, t} = I + U2(x, t), where uz(x, t) -> 0 as t -> 0. Let us prove the necessity. Suppose that u(x, t) —> / as t —>• oo, for any x £ K. In particular, for any r > 0 we have u(x, rqan) —> /,

n -> oo.

According to (4.28),

[exp(-argam) -

Parabolic equations

149

where the sequences v = {vm}2?oo and WT = {wr,m}2°oo are given by

vm — MJmV,

wTtm — exp(-arq'am) - exp(-arqal-m+^).

It is easy to verify that v £ /oo(Z), wr € /i(Z), and wv,m > 0, 00

Y^ wTi7n =

^—^ v~ — oo

lim exp(-orgam) - lim exp(-argam) = 1.

m—*• —oo

m—>oo

Let us consider the Fourier transform (in the sense of harmonic analysis on the group Z): oo

uj^(^i) = \

e~zm'*wT)jTj,

0 < // < 2?r.

J/= — OO

We shall show that for any p. €. [0,2?r) there exists such r > 0 that «£(//) 7^ 0. Suppose the contrary, i.e. that for some /J.Q £ [0,2?r) oo

53 e-inw [exp(-argam) - exp(-Or5a(ro+1>)] = 0

(4.32)

m=—oo

for all r > 0. It is clear that HQ ^ 0. Differentiating (4.32) termwise with respect to T we find that - aqam exp(-ar^aro) = 0. (4.33)

This series is absolutely convergent, uniformly with respect to r £ [ri,r2], 0 < TI < T2 < oo, so that the differentiation was legitimate. Denote the right-hand side of (4.33) by F(T). It is important that in (4.33) (in contrast to (4.32)) one can expand the square brackets and present the series as the difference of two convergent series. After an obvious change of variables we arrive at the representation

where fm = e-*"»Mo(ew> _ l)g«"», Aro = -aqam. The function F(T) extends analytically into the half-plane r + iO, T > 0, 6 € E. For each fixed T > 0 the series

150

Chapter 4

converges uniformly with respect to 0. Therefore F(T + id) is Bohr almost periodic in 6. On the other hand, by (4.33) and the uniqueness theorem for analytic functions, F(T + id) = 0. Now it follows from the Parseval identity for almost periodic functions that fm — 0 for all m € Z, whence Ho — 0, and we have got a contradiction. Thus we have proved that

M e [0, 27r) : t£(A*) = 0} = 0.

(4.34)

r>0

Consider the sequence

,

„

[0,

n 7^ 0.

By Wiener's Tauberian theorem for the group Z (see [55]), it follows from (4.34) that OO

MJnV = (w * * ) „ — > / as n —> oo. The theorem is proved. • Theorem 4.4 can be compared with similar properties of classical parabolic equations [29].

4.4

General uniqueness theorem

Let us consider a more general parabolic equation of the form

uu

— +a0(x,t)(Dau)(x,t)+ b(x,t)u(x,t)= f ( x , t ) ,

x£K,te(Q,T],

(4.35)

0 < ai < a2 < ... < an < a. In this section we assume that the coefficients ak(x, t), k — 0,1,... ,n, are non-negative bounded continuous functions, b(x, t) is a continuous bounded function. Let 0 < 7 < ai (if ai(x,t) = ... = an(x,t) = 0, then we shall assume that 0 < 7 < a).

Theorem 4.5. I f u ( x , t ) is a solution of the equation (4-5) with f ( x , t ) = 0, such that u € 9J17 uniformly with respect to t, and u(x, 0) = 0, then u(x,t) = 0.

Parabolic equations

151

Proof. We may assume that b(x, t) > 0 (otherwise we would be able to consider the equation for u(x,t)ext with an appropriate A). Let us prove that u(x, t) > 0. Suppose the contrary, that is ii(r' >• •£ 0 U,\^L , t'} U j — — M << U

for some a;' 6 K, t' € (0,T]. Denote ; 0, s > r. Fixing a natural number / consider the function

6i(x) = qlu>q-i(\\x\\),

x £ K.

As we know, 61 £ D(K), f6i(x)dx = 1. Let us choose r] > 0 in such a K way that 7 < rj < ai, or if a\(x, t) = ... = an(x,t) = 0, then 'j < rj < a. Denote

The function ^ is locally constant, and i^(x) — ^x^, if ||a;|| > q~l. Further,

z?°^ = (/_„ * st * fn+i)TK(n +1) = rv(7 + i)/,-a+i * *

(4.36)

(see Chapter 2); recall that fs £ T>'(K) (s ^ 0,1) is a distribution, which \\x\\s~l corresponds (if s < 0, after the regularization), to the function ' ; IV(s) also /0 = 5. Similarly,

Da^ = TK(rj + l)fr,-ak+i * Si.

(4.37)

It follows from (4.36) and (4.37) that (Da^)(x), Da*^(x) -> 0 as ||x|| -^ oo, and that Datp, Daki4> are locally constant functions on K. Let

M=

sup x€K,Q
f

{a( V

Choose p > 0 in such a way that x + Tp < 0. Then choose a number a > 0 so small that

p + (Tip(x') < 0 , - o-M > 0.

(4.38) (4.39)

152

Chapter 4

Consider the function v(x, t) — u(x, t) + pt + crtjj(x).

It is seen from (4.38) that v(x',t') < 0, so that inf

x<eK,0
v(x, t) < 0.

Since u(x,Q) — 0, we have v(x,0) = cn/)(z) > 0. As ||x|| -> oo, u(x,t) = o(ip(x)) because u £ 2T7 and 7 < TJ. Thus v(x, t) > 0 if ||x|| is large enough. This implies the existence of such XQ G K, to € (0, T], that

i n f v ( x , t ) = v(x0,t0) < 0. On the other hand,

dv(x,t)

, a0(x,t)(Dav)(x,t)

+ 2^ak(x,t)(Dakv)(x,t)

+ b(x,t)v(x,t)

k=l = p + cr

a0 (x, t) (Da^) (x) +

ak(x,t) (Dak ^) (x)

according to (4.39). However it follows from the hypersingular representation (2.8) for the operator Da that at the point (xo,to) of the global minimum we have (Dav)(xo,t0) < 0, (Dakv)(x0,t0) < 0. It is clear that o

/

i \

also — ,°' < 0. We have come to the contradiction. dt ~ Thus we have proved that u(x, t) > 0. Considering — u(x, t) instead of u(x,t), we find similarly that u(x,t) < 0. Therefore u(x,t) = 0. •

4.5

Fundamental solutions of parabolic equations

4.5.1. Parametrix. In this section we construct a fundamental solution of the Cauchy problem for the general parabolic equation (4.35). We shall assume that a > 1, 0 < a\ < ... < an < a, the coefficients ao(x,t), BI(X, t),..., an(x, t), b(x,t) belong (with respect to a; € K) to the class 9#o uniformly with respect to t € [0,T], and satisfy the Holder condition in t, with an exponent v 6 (0,1], uniformly with respect to x € K. We

Parabolic equations

153

assume also the uniform parabolicity condition ao(x,t) > /j, > 0. Without

loss of generality we may assume that an+i = a(l — u) > an. As in the Euclidean case [29, 97], the first step of the construction of a fundamental solution is to study the parametrized fundamental solution Z(x, t, y, 6) of the Cauchy problem for the equation

where y 6 K and 6 > 0 are parameters. We have (see (4.4) and (4.5))

Z(x, t, y,9) = j

x(-x®

exp(-a0(y, 6)\\t\\") d£,

(4.40)

K

and if x ^ 0, then _-\\m ill.

=l

i

1

-i _ nam y

-i

By Lemma 4.1 and Lemma 4.2 we have the estimates (4.42) (4.43) 1

,

(4.44)

0 < 7 < a,

where the constants do not depend on y,0. As before, we get the identities

Z(x,t,y,6)dx = l,

(4.45)

K

dZ(x,t,y,0)

= -a0(y, 6)

\\t\\ax(-x® exp(-oo(l/,

,

(4.46)

x 7^0, K

(4.47) (4.48)

One more estimate is given by the following lemma.

Chapter 4

154 Lemma 4.5.

dZ(x - y, t, y, 6)

dt

dy

(4.49)

Proof. It follows from (4.45) and (4.46) that

/ K

8Z(x,t,y,9) dt

dx =

(4.50)

Therefore

/ K

8Z(x-y,t,y,6)

d

=

dt

ndZ(x-y,t,y,0) ^ •dy

dZ(x-y,t,x,0)

dt

dt

dy.

K

(4.51) The function —— ' ' belongs to the class in the variable y, and dt its local constancy exponent (which is equal to the one for OQ by virtue of (4.46)) does not depend on the rest of variables. Thus the integral in the right-hand side of (4.51) is actually taken over the set {y 6 K : \\y — x\\ > q~N}, where N does not depend on x, t, 6. Now the inequality (4.49) follows from (4.43). • 4.5.2. Heat potentials. Consider the heat potential

t u(x, t, T) = j dO j Z(x -y,t-6,y, 0)f(y, 9) dy, T

(4.52)

K

with / € yR\ (0 < A < a) uniformly with respect to 0, and / is continuous in (y,0). Note the difference between (4.52) and the heat potential studied in Sect. 4.1 - the integral in (4.52) is not a convolution. In those cases where it is sufficient to use an estimate for Z or its derivatives (see (4.42)(4.44)), the corresponding integrals are actually convolutions. Thus we obtain (as in (4.16)) the estimate

u(x,t,r)\ <

\\x\\x),

(4.53)

which means that u £ 9Jl\. In order to obtain the expressions for derivatives of u(x, t, T) we repeat the arguments of Sect. 4.1, but in a modified way, taking into account

Parabolic equations

155

the dependence of the kernel Z on its third and fourth variables, its local constancy with respect to its third variable, and the identities (4.46), (4.48). As a result we obtain the following identities: )2/

'

[f(v,6) - f ( x , 0 ) ] d y

K

-—\——^— dy;

(4.54)

K

(D^u)(x,t,T) = f dff fz^(x-y,t-0,y,0)f(y,0)dy, J

J

T

K

(4.55) A < 7 < a; * a

(D u)(x,t,T) = I dO j Za(x - y,t - 0,y,0)[f(y,0) T

- f(x,0)]dy

K

t

+ f f(x,0)d0f[Za(x-y,t-9,y,0)-Za(x-y,t-0,x,0)]dy. T

K

(4.56) The formulas (4.54)-(4.56) remain valid also under slightly more general assumptions regarding the function /: it is sufficient to require that / be uniformly locally constant in x £ K, be continuous in (x,t), t € (0,T], and satisfy the bound

\f(x,

t)\ < Ct~p(l + \\x\\x),

0
(4.57)

4.5.3. Cauchy problem. Let us consider the Cauchy problem with the initial condition (4.2) for the equation (4.35). The conditions on / and (f are the same as in Sect. 4.1 (see Theorem 4.1): tp € 9Jl.\, / € 9Jl\ uniformly with respect to t, and / is continuous in (x, t). Here we assume that 0 < A < ai ; if all the coefficients ai , . . . , an vanish identically, then we may assume A < a. As above, we look for a solution u(x,t), which is continuous on Kx [0, T], continuously differentiable in t, and belongs to 971^ uniformly with respect to t.

156

Chapter 4

Theorem 4.6. The Cauchy problem (4-35)-(4-2) possesses a solution

t

u(x, t) =

dr 0

T(x, t, £, r)/(£, r) df + K

T(x, t, £, OMO df,

(4-58)

K

where a fundamental solution r(x,t,£,r), x,££K,Q
(4.59)

-a-l

_ r )l/« + l l a ; _ ( J

,

(4.60)

He Z satisfies the estimates (4-42), (4-43), (444), (4-49). Proof. Following the usual scheme of Levi's method [29, 97], we look for a fundamental solution of the Cauchy problem in the form (4.59), where

t W(x, t, £, r) = j dO j Z(x -r,,t-9,r,, 0)*(r,, 9, £, r) dr,, 0

(4.61)

K

and $ is determined from the integral equation

t $(z, t, £, T) = R(x, t, £, T) +

d6

R(x, t, r,, 0)3>(r,, 0, f, T) dr,,

(4.62)

in which

;,r) -ao(x,t)]Za(x-£,t-T,£,T)

,(x-£,t-T,£,T)- b(x,t)Z(x -£,t-r,$,T).

(4.63)

Using (4.42) and (4.44) we obtain

\R(x,t,£,r)\
(4.64)

Parabolic equations

157

The integral equation (4.62) can be solved by the method of successive approximation: CO

*(ar, t, & T) = ]T R»(x, *, & T), f=i

(4.65)

where RI = R and t

,t^,r)= f d d f In order to prove the convergence of the series (4.65) we need the following lemma. Lemma 4.6. Let

t

J(x, £, t, r) = j(t - nrp/a(» - rY"'a dn T

\(t - tf'a + \\x - r]\\} ^ [(/. - T)1/" + \\r, - £||] ~1~*a dr,,

x K

(4.66)

where 0 < r < t, bi , 62 > 0, p + bi < a, a + 62 < <*• Then

J(x,t,t,T) < C\B(l - ^-^, 1 - -)(t - T)-(P+-+^[

a

\(t _ T )l/a + ||X _ f ||1 -1

L

J

a

£, 1 _ f ± ) (

a

a

t

-

T)

-r^ + lla;-^]'1"61},

(4.67)

where C depends only on &i,&2/ -B is the beta function.

Proof. Let us decompose the domain of integration in (4.66) as HI UII2,

Let also

158

Chapter 4

€ H2 : (t- /z)1/0 + ||x - »?|| <

(t -

n22 = n2 \ n21. NOW (Tit)*K= u n,* and j(x,t,t,T) = £ •/,-*, j,*=l

j,k=l

where /,-£ is the integral over U.jk • Let us use the elementary inequality (u — v)* > u* — v*, where 0 < v < u/2, >f < 1. If {/a,/?} € HH, then (recall that a > 1) /J,-T < (t-r)/2, (t - M)1/" + \\x - },|| = [(t - r) - (/i - r)]1/" + \\(x - & - (T] - Oil

> (t - r)1/* + Hx - ai - [(A* - r)1/" + ||T? -

whence

r

/"
-1-6,

/

(t+r)/2 X

T

After the change of variables t — p = (t — r)v we get 1,62)

[(*-T) 1/a + ||:r-£||]~ i x

' (t - T)~

1/2 a

a r

.,

x (i-r^ + H a r - l l l

-1-1-61

Parabolic equations

159

Next,

(i+r)/2

J

J

r

K

As before,

p + bi , a

0\. a

After similar estimates for J2i and J22 we obtain the inequality (4.67).

Now we return to (4.65). From the estimate (4.64) and Lemma 4.6 we obtain by induction the estimates of iterated kernels:

*•

' k=l

It follows from the Stirling formula that the series (4.65) is convergent and

\*(x, t,S, r)| < C £ [(t - r)1/- + ((a; - £||] " afc ~ 1 .

(4.68)

k=l

Now from (4.42), (4.61), (4.68), and Lemma 4.6 we obtain the estimate (4.60). Denote by ui(x,t) and M 2 (o;,i) the first and second summands in the right-hand side of (4.58). Substituting (4.59) and (4.61) into (4.58) we find that

ui (x, t ) - j dr I Z(x -£,t-T,t, T)/(£, r) d£ 0

K

j Z(x - r), t - 0, r], e)F(n, 6) drj, K

160

Chapter 4

u2(x, t) =

Z(x - £, t, &

+ j d6 j Z(x-rj,t-6, r), 6)G(r,, 6) dr,, where

F(r,, 9) =

dr K

0

It follows from the inequality (4.68) and Lemma 4.3 that

\F(ri,0)\
Rv+l (x + 6,t,£ + 6,r)=

d6 T

T

R(x + 6, t, 77, 0)Rv(n, Q,£ + 6, T) dr] K

K

so that $(x, t, £, T) = $(x + 6,t,£ + 6,r), whence

F(r, + 5,6} = j dr j $(77 + 6,6, £ + S, r)/(£ + 6, r) d£ = F(r,, 6) 0

K

Similarly G(r, + S,8) = G(rj,6), \\6\\ < q~N .

Parabolic equations

161

Thus the potentials in the expressions for ui(x, t) and u-z(x,t) satisfy the conditions, under which the differentiation formulas (4.54)-(4.56) were obtained. If we use those formulas, along with (4.46) and (4.47), we find after some simple transformations that the function u(x, t) E 9Jl\ satisfies the equation (4.35). Let us show that u(x, t) ->• ip(x) as t -» 0. Let

K

Due to (4.59), (4.60), it is sufficient to verify that v(x,t) -> • 0. By virtue of (4.45) we have

(x, t) = j[Z(x - £, t, $, 0) - Z(x - £, t, x, K

Z(x - & t, x, K

Since Z (as a function of its third argument) and

q-N}.

Applying the inequality (4.42) on this set, we see that both integrals tend to zero as t -4 0. • 4.5.4. Non-negative solutions. Let us consider the situation, in which the conditions of Theorems 4.5 and 4.6 hold simultaneously.

Theorem 4.7. If the coefficients ak(x,t) andb(x,t) are nonnegative, then Proof. It is sufficient to show that the solution

K of the Cauchy problem for the equation (4.35) with f ( x , t) = 0 and the initial condition U(X,T) = 0, ip 6 "D(K), is nonnegative. It follows from (4.42), (4.59), (4.60) that u(x,t) -> 0 as

162

Chapter 4

\\x\\ —¥ co. Hence, if u(x,t) fails to be nonnegative, then there exists a negative global minimum point:

u(x0,t0)=

inf

xeK,0
u(x,t)<0. V '

(4.69)

V

'

As in the proof of Theorem 4.5, this can be shown to imply that at the point (xQ,to) all of the terms on the left in (4.35) are nonpositive, and hence are zero. Since a0(xo,t0) > 0 by our uniform parabolicity assumption, we find that in particular (Dau)(xo,t0) = 0, which is possible only if u(x,to) =const, so that u(x, to) = 0. This contradicts (4.69). • The following relation is deduced from Theorem 4.5 in a standard way: r(ar,t,£,T) = jT(x,t,y,a)T(y,o-,£,r)dy,

T < a < t < T.

K

If b(x, t) = 0, then in addition

K Suppose now that the left-hand side of (4.35) is defined for all t > 0, and all the conditions, under which Theorems 4.5 and 4.6 hold, are fulfilled for any T > 0. As in Sect. 4.2, we come to the following probabilistic interpretation of the fundamental solution.

Corollary 4.1. The fundamental solution r(x,t,£,r) is the transition density of a bounded right-continuous strict Markov process without second kind discontinuities. If b(x, t) = Q, then the process does not explode.

4.6

Heat equation on a ball

4.6.1. The process and its generator. Let £«(£) be the Markov process on a local field K constructed in Theorem 4.2. In this section we shall use £a(t) in order to construct a natural Markov process on a ball Br = {x£K : \\x\\
Suppose that £a(0) € Br. Denote by £« (t) the sum of all jumps of the process £ O (T), T € [0, t], whose absolute values exceed qr'. Since £a is right continuous with left limits, £«r (t) is finite a.s., £« (0) = 0. Let us consider the process

Parabolic equations

163

Since the jumps of rja never exceed qr by absolute value, this process remains a.s. in Br (due to the ultra-metric inequality). Such a property is of purely non-Archimedean nature; nothing of this kind can take place for real-valued stochastic processes. Having constructed a process in the ball Br we will now find its generator A". Since the operator — Da is nonlocal we should not expect A" to coincide with — Da on locally constant functions concentrated on Br. However it turns out to be connected with -Da in a remarkably simple way. Below we will often identify a function on Br with its extension by zero onto the whole of K. Let T>(Br) consist of functions from T>(K) supported in Br. Then T>(Br) is dense in Lt(Br), and the operator D" in L-2(Br) defined by restricting Da to T>(Br) and considering (Da(p)(x) only for x € Br, is essentially self-adjoint and positive (see Chapter 3, where this operator was denoted Asr. Denote by A its smallest eigenvalue. Then (see Sect. 3.3) A=

9 1 ~ o^1-^
Theorem 4.8. If rfa(0) = x, and ip 6 T>(BT) then

d

(4.70) t=o

Proof. It is sufficient to consider the case x = 0. By the general result about processes with independent increments [56] the processes r)a(t) and £oT (t) are independent. Thus

for any z £ K. We know that Ex(zf a (i)) = e\p (-t\\z\\a). The Levy measure of the process £Q equals

Since

f [x(zx) - IK (da:) 1 J

K\B \B,

\

J

(see [56]) we find from (4.71) that

f

F

a(*)) =explt [x(zx) - l]7ra(dar) } . (i

164

Chapter 4

This formula is a special case of a result from [32]. The character \ equals 1 on BQ. Hence ~Eix(zrla(t}) = 1 if \\z\\ < q~~r . Suppose that \\z\\ = qk, k > -r + 1. Then

j(x(zx) - l]-Ka(dx) =

I

\x(zx) - iK

Using the formula (1.28) for the Fourier transform of a radial function we find after elementary calculations that - l]-Ka(dx) = -\\z\\a + A, ||z|| > q~r, Br

so that

! ' ' Cr if \\

,„. „ „ . „ [exp(i(A - I I ^ H ) ) ,

(4-72)

= Fij), V € T>(K), then by virtue of (4.72)

whence

I

ip(z)dz-

Nlxr-

I

\\z\\a^(z)dz.

(4.73)

l\*\\>
By the Fourier inversion formula

K

On the other hand, since supp<£> C Br, we find that ip(z) = V'(O) for \\z\\ ^ 1~T • Computing the integrals we obtain that

\\z\\adz

= A

Parabolic equations

165

so that (4.73) takes the form _ t=

°

K

K

4.6.2. Transition density. Consider the Cauchy problem

du(x t} —^-^ + (£>» (x, t) - \u(x, t)=0,

u(x, 0) =
x £ Br, t > 0;

x € -Br,

(4.74)

(4.75)

where tp € V(Br). A maximum principle argument (as in the proof of Theorem 4.5; note that a simple substitution reduces the equation (4.74) to a similar equation without the term Aw ) proves the uniqueness of the solution of (4.74)-(4.75). Thus the fundamental solution Zr(x - y,t) for the problem (4.74)-(4.75) can be identified with the transition density of the process r/a. The next result gives a formula for the transition density; its structure is quite different from the ones appearing in conventional parabolic equations. Note in particular its dependence on the difference x - y, x, y e Br. Theorem 4.9. The solution of the problem (4-74)-(4-75) is given by the formula u(x, t) = j Zr(x - y, t)ip(y) dy,

x € Br, t > 0,

(4.76)

where Zr(x, t) = extZ(x, t) + c(t),

n=0

x e Br, n\

1 -^ q-an-1

and Z(x,t) is given by (4-5). The function Zr is non-negative, and Zr(x,t)dx = 1.

(4.77)

Br

Proof. Let us substitute (4.76) into (4.74) remembering that functions on Br are meant to be extended onto K by zero. If d r ( x ) , x 6 K, is an indicator of the set Br then we may write u = HI + 112, where

Z(x - y, Br

166

Chapter 4

u2(x, t) = c(t)9r(x) J
x € K, t > 0.

Br

Introducing also the notation

wi(x,t) =6T(x) I Z(x-y,t)tp(y)dy, B,

w2(x, <) = [!- 0r(x)] j Z(x - y, t) 0, we obtain that

so that for x € Br

——^—-— + (Dawi) ( x , t ) = - (Daw2) (x,t) 0tj

and r\

/

Ul x

^' dt

,\

+ (D"ui) ( x , t ) - X u i ( x , t ) = -ext (Daw2) ( x , t ) ,

x € Br. If z £ Br, then the ultra-metric property of the absolute value leads to the formula

Substituting this into the expression for Da and calculating the integrals we find that for x 6 Br

-q-a-L

(-1)"^ l-g°" J _ Q—a—an—1

On the other hand, " /

BT


Parabolic equations

167

because Or is an eigenfunction of the operator £>" corresponding to the eigenvalue A (see Sect. 3.3). Now it remains to compute c'(t) and perform a simple series transformation, and we come to the equality (4.74). The fact that the condition (4.75) is fulfilled follows from properties of Z (see Sect. 4.1), since c(0) = 0. The inequality Zr(x, t) > 0 and the identity (4.77) are evident from their probabilistic meaning. However it is not difficult to give direct proofs - the non-negativity may be proved by the maximum principle argument (as in the proof of Theorem 4.7), while (4.77) may be verified by an immediate calculation based on the representation

= J'x(-xz)eXp(-t\\z(\a)dz. K

It is well known (see e.g. [50]) that the unit ball £?Q in the 2-adic field Q2 is homeomorphic to the Cantor set C C [0,1]. Moreover the homeomorphism explicitly written in [50] transforms the Haar measure on BQ into the Cantor staircase distribution on [0,1]. Thus the above results, together

with general facts about transformations of Markov processes [27], lead to the explicit construction of the transition density of a Markov process on C, right continuous with left limits.

4.7

Heat equation on the group of units

Let U be the group of units of a local field K, A\j the operator over U obtained by restricting Da to V(U) and considering (Datp)(x), (p £ T>(U), only for x G U. As we saw in Sect. 3.5, the operator AU is connected with the multiplicative structure of the local field K. In particular, by Theorem 3.12, for any

(U)

(Aw) (x) = !*"£-! / II 1 - yll'"-1 fcteT1) - ¥»(*)] dy + Aov(ar),

u (4.78) „« —1 _ 2(7~1 4- 1

x G U, where A0 = —-——3—j—— is the smallest eigenvalue of the closure

A^on L2(U).

168

Chapter 4

It is natural to consider the Cauchy problem

d"fo*) + (AUU) (x, t) - \0u(x, t)=0,

x € U, t > 0;

(4.79)

t/fc

u(x,0) = tf(x),

x£U,

(4.80)

where <^ € T}(U). As before, we shall understand AUU (where u need not belong to ~D(U)) in the sense of the hypersingular integral in the right-hand side of (4.78). The next theorem gives a fundamental solution for (4.79)-(4.80) (understood in the sense of the multiplicative structure of K and U). As in the previous section, it is obtained by a certain transformation of the fundamental solution (4.5) of Da over K.

Theorem 4.10. The solution of the problem (4-79)-(4-80) is given by the formula

u(x,t)= I Zf(xy~1,t)ip(y)dy, u

x 6 U, t > 0,

(4.81)

where

Zf(x,t)=eXotZ(l-x,t)+c0(t),

c0(t) =

x€U,

9-1

The function Zf is non-negative, and

I Z (x t) dx = 1 u

(4 82)

Proof. Let us substitute (4.81) into (4.79) remembering that functions on U are meant to be extended onto K by zero. If T](X), x G K, is an indicator of the set U, then we may write u = MI +11%, where «i (a;, t) = eXotr)(x) j Z(l-xy~\ t)V(y) dy

u = eXotn(x) j Z(x - y, t)V(y) dy,

Parabolic equations

169

u2(x, t) = cQ(t)r](x)

I (?(y) dy, u

x 6 K, t > 0.

Introducing also the notation wi (x, t) = r](x) I Z(x - y, t)tp(y) dy, u

, t) = [1 - r](x)] I Z(x - y, t}(p(y) dy,

x € K, t > 0,

we obtain that

~ + Da ) wi = - I ~ + Da dt I \dt so that for x € U

dwi(x,t)

dt and dui

^ + (AUU1) (x, t) - XoUl(x, t) = -eAo* (Daw?) (x, t).

By the ultra-metric property of the absolute value W2(x,t) =

I (

if \\x\\ < I , and a al J n / T J'

•*• * —— '

'2V ',*) 'i

' « ( n . \ rfn,

= /<

TL

*

'

~

-f"M/*>ll—&n~~1

if 1 1 a; 1 1 > 1. Remembering that W2(x,t) = 0 if ||a;|| = 1, substituting into (4.78) and calculating integrals we find that for any x € U

(D«w2)(x,t) 1 -

g -«-l

y !7

=1 "—

n!

(1 -

9 -an-l)( g an+a+l

_ 1) '

170

Chapter 4

On the other hand,

a

r\

\

\

f

— + Av - Ao ) w2 ) (x, t) = c'0(t)rj(x) I
u

because 77 is an eigenfunction of the operator AH corresponding to the eigenvalue AQ. Now it remains to compute c'0(t), and after elementary (though somewhat tedious) transformations we come to the equality (4.79). The fact that the condition (4.80) is fulfilled follows from properties of Z since c0(0) = 0. The inequality Z*(x,t) > 0 can be proved by the maximum principle argument just as it was done in the preceding sections. The identity (4.82) is verified by a direct calculation based on the representation

K

The fundamental solution Z» can be interpreted as a transition density of a non-exploding Markov process on U. It follows from the expression for Z*, properties of Z and the general theory of Markov processes that the process is right-continuous, with left limits, without second kind discontinuities.

4.8

On parabolic pseudo-differential equations over Rn

4.8.1. Motivation. It is interesting to compare the above results with those for parabolic pseudo-differential equations over R n . The main difference is the problem of convergence of hypersingular integrals. In the non-Archimedean case one has rich spaces of locally constant functions, on which hyper-singular integral operators exist due to the actual disappearance of singularities. The Euclidean case is much more complicated - rather subtle regularization procedures are necessary to guarantee the convergence. The regularisation depends crucially on the order of an equation. In particular, if the order is greater than 2, the regularized operator has a form preventing the use of the maximum principle argument used in the proof of Theorem 4.7 in order to show the non-negativity of the fundamental solution. Thus the problems of convergence and positivity in the Euclidean case are closely connected.

Parabolic equations

171

We shall consider the Cauchy problem

^g^ + (Au)(x,t) + Y^(Aku)(x,t) m k=i

=f(x,t),

x&Rn,te (0,T],

(4.83) «(x,0) = ¥>(*),

(4.84)

where A,Ai,... ,Am are pseudo-differential operators with the symbols a(x,t,£),ai(x,t,£),... ,am(x,t,£), that is, e.g.

(Au)(x,t) = (27r)~ n / etx'^a(x,t,^u(^,t)d^,

(4.85)

where

The principal symbol 0(0;, t, £) is assumed homogeneous of the order 7 > 1 with respect to the variable £, and elliptic, the symbols ai(x,t, £),..., am(x,t,£) have the orders of homogeneity 7*, 0 < jk < 7, f and ip are bounded continuous functions. The expression (4.85) can be used only for smooth rapidly decreasing functions, and our first task is to reformulate it in terms of hyper-singular integrals. 4.8.2. Hyper-singular integrals. Suppose we have complex-valued bounded functions / € C(Rn), 0 € C(Rn x S™"1). The expression

where a > 0, (A^/) (z) = XI (~"l) ft I . I f(x ~ kh), I is a natural number, k=o V^/ dnti(a) is a normalization constant (its choice will be explained below), is called a hyper-singular integral of the order a with the characteristic fi. We shall also use a restricted hyper-singular integral D^ £ /, which has the same form (4.86) except that the domain of integration is restricted to {h £ E™ : \h\ > e}, e > 0. The theory of hypersingular integrals was developed by Samko (see [130, 131]) for the case of a characteristic independent of x. The results we use below carry over virtually unchanged to the general situation.

172

Chapter 4

Suppose first that the number a is not an integer. The integral (4.86) is absolutely convergent if / > a, and the function / has bounded derivatives up to the order [a] + 1. The absolute convergence over the ball {\h\ < e} is a consequence of the formula

E

(4-87)

where 0 < 6V < 1, / > p., and the usual notation for operations with multiindices is employed (of course, in this section, unlike the rest of the book, D denotes a usual differentiation). The formula (4.87) follows from the Taylor formula. The restricted hyper-singular integral is absolutely convergent if, for example, / is bounded. The above convergence conditions can be easily extended to the case when f ( x ) and its derivatives grow not too rapidly as | a; -> oo. We shall use also conditionally convergent hyper-singular integrals of bounded functions: (Dg/) (x) = lim (Dg ie /) (x),

(4.88)

if the limit (4.88) exists for all x € E". If the characterisic tl(x,a) is even in 2[a/2], by the formula

where this time / + 1 > a. Next, let a be an integer. If a is even, then (D^/) can be denned as before. In this case the hyper-singular integral operator DQ is actually a differential operator of the order a. If a is odd, then for I > a the integral in (4.86) vanishes identically for any function /. In this case D£j can be defined only for an even characteristic fl by the formula (4.89) with / = a. Let / € C7'(K»), |/(*)| < C(l + \x\)~N\ \(D"f)(x)\ \x\ = I, where NI > a, N2 > n. Then

a;) | < C(\ + |z|)-

Parabolic equations

173

Moreover, if ft depends on a parameter t being bounded uniformly with respect to t, then the constant in (4.90) does not depend on t. The same estimate, with a constant independent of e, holds for the restricted hypersingular integral DQ I£ /. Let us consider the function f$(x) = e^'x. Obviously, (A'fc/«) (*) = e^ £(-!)

k=o

whence (D^/?) (20 = 0(2,0^),

(4.91)

where

The function ft is called i/ze symbol of the hyper-singular integral DQ/. The normalization constants dnj(a) are chosen in such a way that the symbol (hence the hyper-singular integral) does not depend on /. Their explicit form can be found in [131]. Here we only note that they do not depend on ft, and dnj(a) > 0 if 0 < a < 2. The symbol 0(or,£) is a homogeneous function of the degree a; for example, if f l ( x , a ) = 1, then Ct(x, £) = |£|a (the corresponding hyper-singular integral operator is often called a Riesz derivative) . The symbol can be represented as f

(4.92)

Note also that the symbol ft is even if and only if the characteristic ft is even. It follows from the formula (4.91) that on functions from the Schwartz space <S(Rn) (and more generally, on smooth functions, which decrease rapidly enough) the operator DQ coincides with the pseudo-differential operator with the symbol ft. Indeed, if / 6 <S(En), then

R"

and by the Fubini theorem X I -— t ^7T)

174

Chapter 4

In order to obtain a representation of a pseudo-differential operator in the form of a hyper-singular operator we shall need some results regarding spherical harmonics [112, 131].

A spherical harmonic of a degree v > 0 is a restriction of a homogeneous harmonic polynomial of of the degree v > 0 to the sphere Sn~1. The set of all spherical harmonics of a degree v > 0 is a finite-dimensional subspace Hv of L2(Sn~l). Let 8V = dim#,,. Let us choose, for each v, an orthonormal basis {Yvll}, // = !,...,(£„. The function Yvti is even if v is even; otherwise it is odd. {YVft} is a complete system in L2(Sn'1). If f £ C2r(Sn~l), then its Fourier coefficients _ vn

r J

——— vn

satisfy the estimate K/il < <7n,PMi/-2r}

where M =

sup

(4.93)

\D*ip(cr). It is also known that

Let 1^M(^) be the symbol of the hyper-singular integral Dy^ /. Then if a is not an integer, or a is an odd integer and v is even, then J.

Kl where

P (n+a\ r /I _

cos • , - . , . , i ———-— (if a is not an integer), COS-2-

Ev>a = (-I)1'/2

(if a is an odd integer).

Let us consider a pseudo-differential operator of the form (4.85), where the symbol a (or, t, £) is continuous and homogeneous in £ of the degree a, and either a is not an integer, or the symbol is even in £ and a is an odd integer. Suppose that

Parabolic equations

175

where N > 2n + a — 1. Expanding the symbol by spherical harmonics,

t/=0 /*=!

we set (4.95)

According to (4.93), \c v f l (x,t)\ < Cv~N+l. Then, since

I

a +v

a?r cos •

or cos

a +v

sin

depending on whether v is even or odd, we find that


n +a +v

r i- v — a

If a is an odd integer, then cVIJi(x,t) = 0 for odd v. Let us use the identity [7] „ / A v — a\ n

* V " 2 ;-

a?r . (v -a \ (in particular, it is equal to cos 2 or to sin -— n 2 \ 2 ) is equal to 1 if a is an odd integer and v is even), and Since

5m

r(3±f±*)

<

(see [7]), we have

\(v,a)

<

which implies, together with (4.94), the estimate


176

Chapter 4

Let us consider the hyper-singular integral operator DQ. Its symbol is

oo a

Sv

/ £\ c

= Kl E E "«(*. *)*W 1^ ) = a(x, t, 0 j/=o/j=i VI? I / (it follows from (4.92) that the termwise transition from the characteristics to the symbols was legitimate). Thus a pseudo-differential operator A can be represented on «S(En) as the hyper-singular integral operator with the characteristic ft. 4.8.3. Parametrix. Suppose that the principal symbol a(x, t, a0 > 0, x € M™, t £ [0,T], \cr\ = 1;

b) a(x, t, a) has N continuous derivatives in a for a ^ 0, and

- a(y, T, a)]| < CW |a: - 2/|A + \t - r\x/

\a ~t-\x\

for all \x\ < N, x,y,a £ W1 (a ^ 0), t,r £ [0,T]. Here A 6 (0,1) is a constant. The parametrix of the problem (4.83)-(4.84) corresponding to "freezing" the symbol at the point x = £, t = 6 is denned by Z(x -y,t-n,t,6) = (2xrn f exp[i(x - y)a - a(t,0,ff)(t

- n)]da,

R»

(4.96) x,ye Rn, t> LL, 9> 0. Let us prove several estimates of oscillatory integrals, which will imply estimates of the function Z and its derivatives. Let v be a non-negative integer, Pv(a) a homogeneous polynomial of the degree v, = f P»(
Parabolic equations

177

Lemma 4.7. // TV > 2n + v + [7], then

\*v(z,t,0)\
(4.97)

where C does not depend on £, #. Proof. Let T?O € C£°(R"), T?O(O-) = 1 as |CT| < 1, 770 (CT) = 0 as | 2. Denote rji(cr) = 1 — r]o(a),

Evidently, $y = $° + $J,. Let us use the representation e~* = l—t+t2h(t), where h(t) = t~'z(e~t — 1 + t). It is easy to see that h(t) has bounded derivatives of all orders on [0,oo). We have

=

770 (d)Pv (o-)eiz"> da-

^ (a)P» (a}a(^ 6,a)e""> da

R"

r!0(a}Pv(a)[a(^ 0,
It is clear that $°'J € 5(Kn). Let av(£,9,a) = Pv(a)a(£,Q,a). Then a y is a homogeneous function of
Let v? € C£°(Kn),
RnRn

= / j]Q(a)av(£,,e,a):tj}(a)da R"

- I av(£,0,a)ip(a) da R™

e^) (z)dz,

178

Chapter 4

where 3^ is the Fourier transform of a,, in the sense of distributions from

S'(Rn), L = -i\z\^ £ k=l

Let us expand the function a,, (£,#,£)) \C\ = 1, by spherical harmonics: oo

61

1=0 n=l

Then for all a e M",

It is known [100] that the Fourier transform of the function Icr!7"1""!^ ( f^T )

equals V

2

(4.98) v ^

'

if / — 7 — z/ ^ 0, —2, —4, . . . (see [40] regarding the interpretation of the homogeneous function (4.98) as an element of <S'(En)). Otherwise (that is possible only for a finite number of values of /) the above Fourier transform is a distribution concentrated at the origin. If \z\ > 1, we have

r

("+^7+1/) Yl"(R)\ , > n „/,_,_„\

' l,l«+7+" f '

/0

^°'

v ; (4.99)

The series (4.99) is a priori convergent in the distribution sense. However by virtue of (4.93), (4.94) it converges uniformly for \z\ = 1, so that for z >1

(4.100) where the constant does not depend on £, ^.

Next, f (LNe^}V(z)dz=(-i)N

J

^ D? [ ,.T^», x\=N

V

Parabolic equations

179

(recall that
)dz R

n

t,6,a)]da I \z\~2N z"eia'z
Z^

J

|H=A

R"

Obviously,

where R(£,B,a) = 0 for 2. The function D%av(£,6,a) is homogeneous in a of the degree j + i/ - N < -n. By the Fubini theorem

dtr

whence for

dz,

> 1

Z* \x\=N

e, a] + R(£, 6, a)} eiff'z da,

R

and due to (4.100)

(4.101) where C does not depend on £, ^. Let us find a bound for 3>°'3(z,£,0), \z\ > 1. Let z = (zi,... ,z n ); consider such a number j that |zj| > z^j, k — 1,... ,n. Then |z| < -^/nlzjl. Integrating NI = n + v + [7] +1 times by parts with respect to the variable

Chapter 4

180
/'

e'er.v

Since rjo(a) = 1 as |<j < 1, all the terms emerging in the process of differentiating, which contain derivatives of r/o, are continuous and have compact supports. Next, computing the derivative

with the use of the Leibnitz formula and the Faa di Bruno formula for higher derivatives of a superposition of functions [47], it is not hard to calculate that the maximal order of singularity at the point a = 0 does not exceed n + [7] + 1 — 87 < n. Therefore for \z\ > 1 we have

that is [$£(*,£,0)| < C\z\-n-~<-v. The function $^(z,£,0) is estimated in the same way as $° > 3 (z,£,8), due to the fact that T?I(
so that |$,,(z,£,0)| < C\z\-n~~<-v,

—v

> 1. In order to prove (4.97), it

remains to notice that

as z\ < 1. • Let <3a( r i (J ) be a homogeneous function in a of the degree a > 0 depending on a parameter r of an arbitrary nature, which has, as a ^ 0, N continuous derivatives in
whenever x\ < N, a ^ 0, for all values of r. Let us consider the function

* a (z, f, 0, r) = I Qa(r, a) exp{iz • a - a(£, 9, a)} da.

Parabolic equations

181

Lemma 4.8. IfN>2n + [a] + [7] + I , then *a(z,t,9,r)\ < CB(r}(l +

(4.102)

where C does not depend on z,£,6,r. The proof consists essentially of repetition of the arguments from the proof of Lemma 4.7. The only new point that appears is the following. In the decomposition of the integral $>a the term

*°'1(0,r) = / rio(a)

(in contrast to the function <J>°'1 from the proof of Lemma 4.7) need not belong to <S(lRn). The techniques of Lemma 4.7 gives the estimate

\z\>l, which leads to the inequality (4.102).

Let us consider the function = I

• a - o(£, 6, a)} da.

Lemma 4.9. IfN>2n + [7], then for any z,£i,& e 1™, B > 0,

The proof is similar to that of Lemma 4.7. Some technical differences are caused by the necessity to obtain estimates of expressions of the form

in two situations: for h(s) = e s, \a\ > 1, and for a function h about which we know only that it has bounded derivatives of any order, for \a < 2.

Chapter 4

182

By the Faa di Bruno formula ~

da]

flo? mi

da]

>,

(4.103)

where the summation is over all tuples (mi, . . . , m M ) of natural numbers, such that mi 4- 2m2 + • • • + /^m,, = I while JJL varies from 1 to /; here k — mi + • • • + m/j,. Let us use the elementary formulas

' v ~ l bv - (b'Y = (b- b')(b Each summand in (4.103) is a sum of expressions containing one of the differences hW(a(£i,6,
dcrj

multiplied by a bounded function and by a homogeneous function of the

degree mi(7 — 1) + 7712(7 — 2) + • • • + mv(^ — v) = /yk — l (in the first case), or 7& — / — 771^(7 — v) (in the second case). Further, a(6,0,(7)) -

Thus for \a\ < 2

da]

183

Parabolic equations for h(s) = e s , \a > 1, we get

CTJ

6

With these estimates taken into account the proof of Lemma 4.9 goes just like that of Lemma 4.7. • The next lemma can be proved in a similar way.

Lemma 4.10. Assume the conditions of Lemma 4-8. Then

where C does not depend on £i,&,d,r. Under the above conditions a), b) we obtain from Lemmas 4.7 - 4.10 the following estimates of the parametrized fundamental solution Z (uniformly with respect to the parameters): - n) (t -

\x- y\]

-"-7-1*1

(4.104) < N - In - [7],

(4.105)

\x- y\] ""~7 ;

< C(t f\

f\

-

-

(4.106)

-n-2-y (4.107)

We have the equation fz(x-y,t-fJL,t,6)dy

=

", t > », 0 > 0,

(4.108)

Chapter 4

184

which follows from (4.96) and the Fourier inversion. Using (4.105) we get from (4.108) that

(4.109) The inequality

/

dZ — (x-y,t-ij.,y,n)dy

(4.HO)

4.8.4. Heat potential. Let us consider the heat potential

u(x, t,r) = / dp, I Z(x -y,t- n,y, fi)f(y,p.) dy. r

(4.111)

R»

We assume the above conditions a), b) with N > In + 2(7] + 1. Suppose

also that

\f(yiP-}\ < C(n — T)~P,

\f(x,fj,) — f ( y , n ) \ < C\x — y\x(^ — T)~P,

where 0 < / 9 < 1 , 0 < A < 1 . The integral in (4.111) is absolutely convergent due to (4.104). It also follows from (4.104) that the function u(x, i, r) has continuous derivatives in x of any order < 7, and they can be obtained by differentiating under the sign of integral. Just as in the non-Archimedean case (see Sect. 4.1 and 4.5) we obtain fjiii \ 3* i~ TI formulas for ——-^—-— and (D^u) (x,t, T), a < 7, which are similar to (4.18) and (4.19). Note only that the calculation of (D^it) (x,t, T) is based on the inequalities

(4.112) 1) if \h\ <(t- ^)1/7, and

-n-7

< c(t j/=0

185

Parabolic equations

if |ft| > (t — /i)1/7, both of which are consequences of (4.87) and (4.104). __ The case a = 7 is much more complicated. Assume that the symbol ft(x,ff) has N > In + 2[7] + 1 continuous derivatives in a ^ 0, and that

Let also fi(x, cr) 7^ 0 if a ^ 0. If 7 is an integer, and the symbol fl(x,cr) is not a polynomial in a (which is possible only for an odd 7 and an even characteristic ft), we assume in addition that in the expansion

i/=0 ^1=1

= 0 if 7 = n + 1v + 2k, k = 0,1,2,.... We shall not consider the case when ft is a polynomial in a because it is covered by the classical theory of parabolic differential equations.

Lemma 4.11. Under the above conditions the hyper-singular integral D^u exists in the sense of the conditional convergence (4-88), and

t

(D^w) (x, t, T) = / d/j, I Zn(x, x-y,t-n,y, n)[f(y,») - f ( x , fj,)] dy T

R»

t

+

f(x,/j,)d(j, T

I [Za(x,x-y,t- fj,,y,fj,) - Z^(x,x-y,t-fi,x,p,)]

dy,

R"

(4.114)

where ZQ, = D^Z, that is Za(x,z,t,$,0) = (27r)-" [ f l ( x , c r ) e x p [ i z - a - a ( $ , 0 , ( 7 ) t ] d < T , R"

and £l(x,a) is the symbol of the hyper-singular integral operator DQ.

Proof. Denote t-s

us(x,t,r) = I dn I Z(x-y,t-n,y,p,)f(y,fj,)dy,

0<s
186

Chapter 4

Using (4.112) and (4.113) we prove the absolute convergence of the hypersingular integral DQ«S, and the formula t-s

(D^us)(x,t,T)=

I d\n I Z a ( x , x - y , t - f i , y , f j , ) f ( y , n ) d y . T

R"

We shall need the following properties of the function ZQ :

n(x z t £ 6}dz = 0'

(4 115)

\Zn(x,x - y,t — n,2/,j«)| < C \(t - /^)1//7 + \x — y\\ L

J

;

(4.116)

\Zfi(x,x-y,t- fj,,y,n) - Z f t ( x , x - y , t - f j , , x , f j , ) \ < C\x - y\\t - A,) [(t - tfl~i + \x- y\] ~"~ ^ . (4.117) The formula (4.115) is a consequence of (4.108); we use (4.112), (4.113), and the Fubini theorem. The inequality (4.116) follows from Lemma 4.8, while (4.117) is a consequence of Lemma 4.10. According to (4.115) t-s

(D^us) (x, t,r)= I d f i l ZQ(x, x-y,t-fi,y, (t)[f(y, p.) - f ( x , //)] dy T

R"

t-s

+ / f(x,fj,)dfj, T

[Zn(x,x-y,t-(j,,y,n) R"

-Za(x,x-y,t-n,x,fjt)]dy,

(4.118)

Let $(x, t, T) be the function in the right-hand side of (4.114) (the integrals converge due to (4.116), (4.117)). Taking into account (4.116), (4.117) we see from (4.118) that uniformly with respect to a; 6 R™

lim (D^ Us ) (x, t, T) = $(x, t, r).

(4.119)

S—^U

Let us consider the restricted hyper-singular integral t-s

D^ j£ u s j (x,t,r) = I d/j, I Za,e(x,x-y,t-n,y,n)f(y,n)dy,

(4.120)

Parabolic equations

187

where e > 0, ZQ!S = D^ £Z. Our immediate goal is to obtain an integral representation expressing DQ eus in terms of DQU S . We introduce the constant characteristic n^(cr) = fi(£, a) depending on £ € E™ as a parameter. Let fi^ be the corresponding symbol. Fixing the rest of the variables we denote F(x) = Z(x — y,t — ^,y,fJ.) and express F via D^ F. Since the

Fourier transform of the function G^ = DQ F equals O^(C)F(C), we find that

F(x) = (27r)-n

where G x ( ^ ) = D F (x + z). Let us expand the homogeneous function 1/Q^ in a series of spherical harmonics:

1

——

X

For each fixed £ € Rn the series converges uniformly with respect to £, so that

ic Rr,

The Fourier transform (in the sense of the space <S'(K")) of the distribution |C|-71W (Ij) equals .-n/2o»-^ r ((» + ^ ~ 7)/2) .

(see [100]), where Yvlt(a) = Yvlt(—ar). Strictly speaking, we cannot use this to transform the integral in (4.121), since Gx£ ^ «S(En). However if n/2 < Re A < n, then the formula

""

A r fdz z

(41221 (4 }

Chapter 4

188

is nevertheless valid, since in this case the function

can be

represented as a sum of functions from Ia(R") and 1/2 (K n ). An explicit form of the function Gx^ and the estimate for the function GXy£ (see [100]) enable us to perform in the standard way [40, 100] the analytic continuation in A and to prove (4.122) for A = 7 ^ n + v + 2k, k = 0, 1, 2, . . . . Denote

i/=0//=l

The series converges uniformly with respect to z. Moreover, it follows from (4.93), (4.94) and the inequality -IH

<

V/J,

that the function H(£, z) is smooth in z for z ^ 0: H(£, •) € Now

F(x) = (1*)-" j H&z) (D R"

whence

(0 dc \h\>

tot

TT7 \ dh

\h\>e

Writing out the expression for the difference A^ in detail and making the substitution h — \£\y, C= erlt after elementary transformations we get

(a;) =

(4.123)

Parabolic equations

189

where

dy,

ffifcer)

The function K(£, rj) is integrable with respect to rj, and = l.

(4.124)

For the case 7 < n this was proved by Samko [130] in connection with the inversion problem of generalized Riesz potentials. The proof in the general case is a word-for word repetition of the arguments in [130]. Setting £ = x in (4.123) and returning to the previous notation we find that

Zn,e(x,x-y,t-fj,,y,fj,)

= I K(x,T))Za(x,x-sr]-y,t-

v,y,tJ.)dr).

R"

Substituting in (4.120) we get that £ I£US (x, t,r) =

K(x, T,) (D£u s ) (x - er,, t, r) drj.

(4.125)

As s -> 0, we have that us(x,t,T) -> u(x,t,r), unfiformly with respect to x e M n . Hence for s ->• 0

(x,t,r) —> D^ jE « (x,t,r). Using (4.119) we pass to the limit in (4.125) as s ->• 0. Then

(D^ £ W) (x, t, T) = j K(x, T])$(X - erj, t, r) drj.

(4.126)

R"

It follows from the uniform convergence (4.119) that the function $(x,t, T) is continuous in x. Now from (4.126), (4.124), and the dominated convergence theorem we get the existence of the limit (D£«) (x, t, T) = lim (D£ )£ w) (x, t, T) = $(x, t,r).

•

190

Chapter 4

4.8.5. Cauchy problem. Now we are ready to consider the Cauchy problem (4.83), (4.84). Let us list the conditions imposed upon the symbols of pseudo-differential operators from (4.83). As before, the principal symbol a(x, t, £) is a homogeneous function with respect to £ of the degree 7 > 1. The symbols a,k(x, t, £) are homogeneous in £ with the degrees 7*, 0 < 7^ < 7. Fixing a natural number N > In + 2[y] + 1 we assume the conditions

a), b) from Sect. 4.8.3 with regard to the principal symbol. Other symbols should be such that if £ ^ 0, x\ < N, x,y E 1", t,r e [0,T], then

where C does not depend on x, y, £, t, T. Some additional requirements (needed in order to represent pseudodifferential operators as hyper-singular ones) must be imposed on the symbols of an integer order. Namely, a symbol of an even order is assumed to be a polynomial in £, and a symbol of an odd order is assumed to be either a polynomial or an even function of £. Finally, if 7 is an odd integer, we assume that in the expansion of the function [a(x, t, £)]-1, |£| =!, in spherical harmonics the coefficients at YIV,\I with 7 = n + 2v + 2k, k = 0,1,2,..., are equal to zero. This condition is automatically satisfied if 7 < n, or if n is even. The initial function (f> is assumed to be bounded and continuous, while the function / is bounded, jointly continuous, and

\f(x,t)-f(y,t)\
By a solution of the problem (4.83), (4.84) we mean a bounded function u(x,t), jointly continuous on E™ x [0,T], which satisfies (4.84), and satisfies the equation (4.83) with A and Ak replaced by the appropriate conditionally convergent hyper-singular integral operators DQ and DQ,., or by differential operators understood in the classical sense, if corresponding symbols are polynomials in £.

Theorem 4.11. The Cauchy problem (4-83)-(4-84) possesses a solution

t u(x, t) = I dr I r(ar, *, f, r)/(£, T) d£ + j T(x, t, & 0

R»

R»

Parabolic equations

191

where a fundamental solution F(x, t, £, r) ; a;, £ € R", 0 < r < f < T, «'s o/ ifee /orm r(x, i, £, r) = £ ( * - £ , * - r, £, r) + W(z, t, £, r),

ra+1

_(j_

Im+l = 7 - A,

while Z satisfies the estimates (4-104) ~ (4-107). The proof is analogous to that of Theorem 4.6. Note in particular that Lemma 4.6 carries over to this case virtually unchanged. • Analogs of Theorem 4.5 and 4.7 also hold in this situation (see [74]). In particular, we have the following property, which is basic for the probabilistic interpretation of the equation (4.83).

Theorem 4.12. If j < 2, and the characteristics fl and Qfc are even and non-negative, then T(x, t, f, T) > 0 for all x, £ e K n , t > T.

4.9

Comments

The formula (4.5) for the heat kernel, as well as the fact of its positivity, were found independently by several authors; see [52, 60, 76, 53, 154, 16, 149]. A systematic treatment of the Cauchy problem (4.1)-(4.2) arid more general parabolic equations over Qp was given by the author [76]. Our exposition of the theory of parabolic pseudo-differential equations over R n follows [74], where references to some earlier papers (by Eidelman, Drin', Fedoryuk and others) can be found. For a generalization to some equations on vector spaces over local fields and division rings see [149]; that paper, as well as [60, 16], contains various versions of the Feynman-Kac formula. An extensive exposition of the potential theory based on the heat kernel (4.5) was given by Haran [53]. The results of Sect. 4.6 and 4.7 regarding heat equations over balls and the group of units are taken from [83, 85].

Chapter 5

Construction of Processes 5.1

Notations and preliminaries

Let £a (t) be the non-Archimedean analog of the symmetric stable process introduced in Sect. 4.2. It is a stochastic process with independent increments on a local field K. We fix the origin as the starting point of £a. As we know, £a is stochastically continuous, and we assume its paths to be almost surely right-continuous with finite left limits. If F is a Borel subset of K \ {0}, denote by qa(t, F) the number of jumps of the process £a belonging to F, on the interval [0,t). If F is contained in the complement K \ Bn of a ball Bn = {x € K : \\x\\ < qn] (n 6 Z) then qa(t, F) < oo. For a fixed F qa(t,T) is a Poisson process with the parameter t-Ka(T) (see [56]), that is

•DS — 1} j\ — i""«v 1 -/J ~ - t K °a (v i )>, n 1+ r\ = "{Qa(t,i) — ——7]——e

i; _— n0 , i1 ,o2 , . . . .

As before, 7ra is the Levy measure,

\x\\-a-ldx. For any fixed t qa(t,T) is a random non-negative countably additive integer-valued measure on the Borel cr-algebra B(K \ {0}). It may also be interpreted as a finitely additive K-valued random measure. If FI nF2 = 0, then the random variables qa(t,FI) and qa(t,Fj) are independent. The characteristic function can be easily calculated:

EX(zqa(t,T)) = exp(ir(t,r)\x(z) - 1]) , z € K. 193

194

Chapter 5

For an interval A = [£', t") C [0, T] we shall write g a (A, F) = ga(i", F) g a (i',F). The random variables g a (A, F) and ga(i" — i',F) have the same distribution. We also use the notation

Tra(dt,dx) = ^ ~^_1\\x\\-a-1dtdx , 7r a (A,F) = j j Ka(dt,dx). A r The theory of integration of K- valued functions on a compact set S C K with respect to a /^-valued measure possesses some features different from the conventional integration theory (see [71, 134, 127]). It is sufficient for our purposes to consider the case of an integer- valued finitely additive measure r(dx). In this case the integral / f(x)r(dx) exists in the Riemann E sense for every continuous function / : £ —>• K and

PII< sup ii/^n.

(s.i)

Note that the measure need not be countably additive (with respect to the topology of K). Below we shall use several Banach spaces over the field K consisting of K- valued functions. The space C(E, K) consists of all continuous AT- valued functions defined on a compact set S; it is endowed with the supremum

norm. If (Q,£, /u) is a measurable space,we define L^(fl,K), 7 > 1, as the set of all functions / : £1 -t K measurable with respect to the
and the Borel cr-algebra B(K ) such that

Completeness of this space can be proved in a standard way (for the properties of measurable functions with values in a general metric space see [113]). If X is a Banach space over K, then D ( [ 0 , T ] , X ) consists of functions on the interval [0,T], with values in X, which are right-continuous and have left limits in X. Well-known properties of the space D summarized in [15, 105] carry over to this situation (in fact they are valid for an arbitrary metric space X). Note that functions from -D([0, T],K) are constant between their jumps; that follows from total disconnectedness of K.

Construction of processes

5.2

195

Stochastic integrals of deterministic functions

Consider a function oo

ak(t)uk ,

f ( t , u) =

0
(5.2)

k=l

where ak € D([Q,T],K), sup \\ak(t)\\l/k =0.

lim

(5.3)

The condition (5.3) guarantees convergence of the series (5.2) for every u € K, uniformly with respect to t € [0,T] (see [71, 134]). In fact, the convergence is uniform on each sphere Sn = {x € K : \\x\\ = qn}, n € Z. Thus / € D([0, T], C(Sn, K)), for each n 6 Z. Our task is to define the stochastic integral T = j j f(t,u)qa(dt,du).

(5.4)

0 K

We assume that K is a local field of characteristic zero, that is a finite extension of Qp. Let us begin with the definition of integrals T In= f ff(t,u)qa(dt,du), 0

neZ.

(5.5)

Sn

Considering / as a vector-function from D([0, T], C(Sn, K)) we can approximate it uniformly by step vector-functions fm(t,u) (see [15]). The function fm corresponds to such a partition

by non-intersecting intervals that the oscillation of the vector-function / on each of the subintervals does not exceed em, where em —)• 0 for m —^ oo. The vector-function fm is constant on A^ . fm(t,u) = fm,i(u), u £ Sn (it can be obtained by fixing the value of f ( t , u) at some point from AJ ), and we may set T fm(t,u)q(dt,du) d^f

Chapter 5

196

where the integrals are understood as in Sect. 5.1. It follows from the estimate (5.1) and the ultra-metric inequality that the sequence {/n)TO}^_1 is fundamental in K almost surely, and its limit does not depend on the choice of the approximations {fm}. We define the integral (5.5) as /„ = lim Inm. m->oo

Using (5.1) again we obtain with probability 1 the inequality Oil-

0
(5-6)

Now we define the stochastic integral (5.4) by the series (5.7)

Theorem 5.1. The series (5.7) is convergent in probability. Proof. By virtue of the ultra-metric inequality it is sufficient to prove that /„ —> 0 in probability when n -> ±00. If n < 0 then according to (5.3) and (5.6) we have with probability 1 \\In\\ <supqnk sup \\ak(t)\\ < Cqn k>l

0
(as before, we denote by C various positive constants) so that almost surely In -> 0 when n -> —oo. Let u € 5n, n > 1, ||u|| < 1. Then fc-i i/=0

< sup t,k,v

sup t,k

1/ n = g- n |H|supf||o fc (*)|| *? Jl*. *,* L

Using (5.3) we conclude that there exists such a natural number l(ri) that

as soon as

<

Construction of processes

197

Let N(n) > 0 be such an integer that sup

o
\\f(t,u)\\
We can write the canonical representation of the element f(t,u) € K as follows:

f(t,u)=n(t,u),

(5.8)

where

n

i,U

= {~

£j = £j (t, u) € S (as before, 5 is a complete set of representatives in the ring of integers O C K of classes from the residue field O/P). Evidently, ll^n(*,«)ll<(7~n,

«€Sn.

(5-9)

n

If u — fi~ (u0+ui(3 + - • •) is the canonical representation of u, then the function
[/(«, u + v)- f ( t , u)} + [f(t, u) -

Vn(t,

u)]||

't,u + v)\\, \\f(t,u + v) - /(t,u)||, ||^n(*,«)||} < q-

On the other hand, N(n)+n-l

3=0

If (pn (t, u + v) ^ ifn (t, u) then

and we come to a contradiction. It is clear that tpn, ipn € D([0, T], C(Sn, K)). We may write In = I'n+I^ where T I

'n= I I Vn(t,u)qa(dt,du); 0 Sn

198

Chapter 5

i/>n(t,u)qa(dt,du).

We have ||/^|| < q~n -> 0 almost surely due to (5.6), (5.9). It remains to consider I'n. Since / € D([0, T], C(Sn, K)), each of the functions £,- may have only a finite number of jumps with respect to the variable t, and the jump points do not depend on u. The same conclusion is valid for the function
E £••• E , «o

where the intervals A^ , i — 1, . . . ,mn, form a partition of the interval [0,T], t^ € A|"\ the set J3(w0, «i, • • -,«/(«)) consists of all elements u e 5n with fixed coefficients UQ, . . . ,u;( n ) in the canonical representation

3=0

Now we get with probability 1 the inequality II T< II s

II nil — qN(n) max

max

I A(«) n^j^^X-

||

whence for any e > 0

p{ii/;n> £ }<E E E - - - E . (5.10)

By our assumption, the field K is a finite extension of Qp. Let M = degJf/Qp. Since q a ( - , - ) is integer-valued, we have ||g a (-,-)l = \
Construction of processes

199

ramification index, / is the index of inertia of the extension K/QP. We rewrite the estimate (5.10) as __

£ £••• £ so that Wtn

___

___

___

£ E £••• E *=1 «o€S,«o£.P «i€S

«i( n )SS Nl(n)

,

(5.11)

where Ni(n) = [e~lN(n)} + 1. Our aim is to prove that P{||/4II > £} -> 0 as n —>• oo. It is sufficient to consider e = p~Mv ', v € Z, v > 0. Using the notation a \ b if a is a divisor of 6,and a \ b in the opposite case, we find that

3=0

3=0

Denoting by |A| the length of an interval A and using the formula for distribution of the Poisson random measure we obtain that

——'<»> < i1 -exp^ exp (-c|A<»> S c|A4 g,

200

Chapter 5

where c > 0. We have used the fact that due to invariance of the measure

/ du

Substituting into (5.11) we come to the estimate

where 5 (D =

^

_exp _

It follows from the inequality e~x > 1 — x, x > 0, that mn 1

< eg """

IA i n) I = eg1"""?1 -> 0 ,

t<\\

n -> oo.

The estimate for Sn is based on the elementary inequality xke~x < kke~k , re > 0 , k > 1.

We obtain that exp -

Construction of processes

201

whence (2)

l-cm V~* (JP^1

j=l By the Stirling formula

so that

-8/2-X>,

-

n-+oo.

j=i Now we can use our stochastic integral to obtain a representation for the process £Q. Note that a similar representation for real-valued processes is not available; one can construct only a representation for parts of the processes "separated from the origin"; see [142]. We shall write £ K j] if the processes £, 77 have the same finite dimensional distributions. If the function / does not depend on t we write f f(u)qa(t, du) instead of

t f f f(u)qa(ds,du). oK

Theorem 5.2. The process £Q admits a representation

£a(t) w / uq(t,du) , K

0 < t < oo.

Proof. Since £a (t) and / uq(t, du) are both processes with independent increments it is sufficient to prove coincidence, for every fixed t, of their characteristic functions. Thus we have to prove that for any 77 6 K

uqa(t,du)

K

.

(5.12)

)

The left-hand side of (5.12) is equal to exp(-£||7?||a). By Theorem 5.1

I r

\

\ K

J

N

I f

\

\ Sn

J

EX h? / uqa(t,du) = Nlim TT E X \ T } uqa(t,du) . \ J I ~*°° n J-LNA: I J / =~

202

Chapter 5

Approximating the integral over Sn by Riemann sums, using independence properties of qa and the expression for the characteristic function we find that

>, ? /t uqa(t,du) \= exp
I

I J

/

\ sn

I

(sn

)

whence

f

f

\

\f

rj I uqa(t,du) J = exp < / J I I J \ K / {K .

= exP(-i||7?|r).

5.3

Stochastic integrals of random functions

5.3.1. The main lemma. Our construction of the stochastic integral will be based on the following general fact.

Lemma 5.1. Let C be a non-negative integer-valued random variable having the Poisson distribution with the mean value a > 0: _„&*

Then for any prime number p, and any 7 £ [1, oo), E|C|? < ——a.

(5-13)

Proof. The inequality |C|P < p~n (n £ Z) holds if and only if £ = mpn+l, m e {0,1,...}. Therefore

- P~n} < P{|CIP > P~n] = 1 -

P{C = mpn+1}

Construction of processes

203

so that

Considering the function

n=0

m=0

we see that y>(0) = 0,

n l ^ v(mp + )\' ^

.n=0

m=0 oo n=0

oo «-!"» ""

Tnp""*"1 —1 ~

^ v(mp™+1 - 1)!.

»T - 1

m=l

e x

oo

oo

i.

~

< ~ E^~ " E ^ ~ ~^r~ = ° n=0

7

k=0

/y«K


'

which results in (5.13). • Returning to our stable-like process £a(t) with values in a local field K, a finite extension of Qp, we obtain the following estimate.

Corollary 5.1. Let A be an interval on the real axis, F C K a Borel set. For any 7 € [1, oo)

E||<7a(A,r)||7 < (77,A-7ra(A,r),

C-y,K > 0.

(5-14)

5.3.2. Definition and properties. Let the probability space (Q, J7, P) be equipped with a right-continuous increasing family Tt (0 < t < T) of sub-cr-algebras of T containing all P-zero sets. Suppose that for any t £ [0, T] and any compact set S C -?^\{0} the random variable qa(t, S) is ^-measurable. We assume also that the random variables g a (si,Si) , ... , g a (s n) S n ), where s i , . . . ,sn €. [t,T], Si,... ,S n are compact subsets of K \ {0}, are jointly independent of any event from TtWe call a random function f ( t , u ) defined on [0, T] x K, with values in K, a random step function if there exists a non-random partition 0 =

204

Chapter 5

t0 < ti < ... < tn = T and non-random Borel sets FI , . . . , Tm C K, Uj=i Fj = K, such that f ( t , u) takes a constant (random) value on every set [tk,tk+i) x TJ , k = 0,1,... ,n-l; j = 1,... ,m. Denote by M(Ft) the set of all ^-measurable random step functions / such that f(t,u) = 0 for u € Bk where k £ Z depends on /. For any function / € M(ft) we define the stochastic integral // as J

n—1 m

= / / f(t,u)qa(dt,du)

=

0 K

k=0j=l

where Uj € Fj are arbitrary points. It follows from the inequality (5.14) and the independence assumptions that EH//IP <

0 K

Denote by M7(J"t) the completion of M(ft)

with respect to the norm

T 117 -

J

We have proved that / >-> I/ is a linear continuous operator from to L7(Q, K) defined on a dense linear manifold M(3:t). Its extension If by continuity onto the whole space M7(^i) will be called a stochastic integral, now defined for any / £ M7(J7(). We have the estimate T

EII//IP < C^K j IE\\f(t,u)\\^a(dt,du).

(5.15)

0 K

Let us describe explicitly a subset of M7(jFi).

Proposition 5.1. Suppose that a random function f(t,u), t € [0, T], u 6 .ftT, with values in K, satisfies the following conditions:

1) f ( t , u ) is ft-measurable for any t,u;

Construction of processes

205

%) f e ^([0, T], C(S, £ 7 (fi, AT))) /or any compact set Z C K \ {0}; 3) ut-t f(t,u) is a continuous mapping from K to L^(fl,K), uniformly with respect to t 6 [0,T];

< oo.

Proof. It is sufficient to consider functions f(t,u) vanishing for qm and for ||u|| > qN , m < N. Denote : qm+1 < \\u\\

It follows from the condition 2 (see [15]) that, given e > 0, there exists a partition 0 = to < t\ < • • • < tr = T for which

sup

sup E||/(s,«)-/(*,u)|r<e,

t = 0 , l , . . . , r - l . (5.16)

ti
Using the condition 3 we can cover Bm^ by non-intersecting balls Vi , . . . ,Vi in such a way that sup

sup E||/(t,«)-/M)|p<e,

j = !,...,/.

(5-17)

t€[0,T] u,t)£Fj

Choose arbitrary points Uj € Vj , j — 1, . . . , /, and set

t (f \- //(*<-i. «j)» if « e [«i-i,*t), « e ^f *' t t ) ~\0, ifutBm,N.

M

It follows from the condition 1 that /e e M(.Ft), and from (5.16), (5.17) that

\\f-fe\\l<2ej

j

7ra(dt,

0 B ,

The right-hand side becomes arbitrarily small when e —> 0.

5.4

•

Stochastic differential equations

5.4.1. Existence and uniqueness. In this section we consider an equation

t

x(t) =x0+

I F(s,x(s),u)qa(ds,du) 0 K

,

0
(5.18)

206

Chapter 5

with a random K-valued function F; a more special case leading to Markov processes will be discussed below. Our presentation follows the general lines of [141]. However there are some features here different from the stochastic analysis in the real domain. First of all our equation does not contain diffusion terms (since there are no continuous processes on a totally disconnected space K ) or drift terms (since one cannot integrate a K-valued function with respect to the Lebesgue measure). We systematically consider stochastic processes (assumed to be continuous in the mean) as functions with values in Ly(£l,K). Thus we are unable to deal with individual paths; we shall not distinguish between stochastically equivalent processes, and the equality sign in the equation (5.18) means in fact stochastic equivalence. The equation (5.18) will be considered under the following assumptions. (Ai) The random variable XQ with values in K is Po-measurable and

EIMP < oo.

(A 2 ) For each x, u € K, t 6 [0,T], the function F(t,x,u) is Ttmeasurable. (A 3 ) F(-,x,u) € D([Q,T],Cb(K x Sn,Ly(fl,K))) , for every n € Z;

here C(, is the space of bounded continuous vector functions with sup norm. (A4) The mapping u >-> F(t,z,u) from K to L j ( f l , K ) is continuous,uniformly with respect to t £ [0,T], x € K.

(A 5 )

/ / E [sup \\F(t,x,U)\\T\ \\u\\-"-1 du < oo.

(A 6 )

For each n € Z there exists such a constant Ln > 0 that almost

surely for all t e [0, T], x, y € K, u 6 Sn

\\F(t,x,u)-F(t,y,u)\\
(A 7 )

E

n= —oo

£nT"(a+1) < oo.

Denote by X the Banach space of (classes of stochastically equivalent) processes x(t), t E [0,T], with values in K, ^-measurable for each t, continuous as functions on [0,T] with values in L ^ ( f l , K ) , equipped with the norm ||a;||x = sup 0
Theorem 5.3. Under the assumptions (Ai) — (Ar) the equation (5.18) possesses a unique solution x € X.

Construction of processes

207

Proof. Consider the operator t (&x)(t) = x0 + j f F(s,x(s),u)qa(ds,du)

,

x £ X.

(5.19)

0 K

The stochastic integral in (5.19) exists by Proposition 5.1 - its condition 1 follows from (A 2 ) and the inclusion x € X, the condition 2 is a consequence of (As) and (As), and the conditions 3 and 4 follow from (A
/ / F(s,x(s),u)qa(ds,du) t K t" 1


Thus (Ai),(A 2 ) and (A 5 ) guarantee that &x £ X. If xi , x2 £ X then

t < C^K j j E||F(s, xi (a), u) - F(s, x2(s),u)\\''iTa(ds, du) 0 K *

oo

.

-C I S inE||ari(s) - a;2(s)|p / Hull- 0 - 1 duds =-

with some fi > 0; we have used (5.15), (A6) and (Ay). Iterating we find that ^ nntn-1.. a;2||x, whence

(5.20)

208

Chapter 5

so that &n is a contraction for sufficiently large n > HQ.

Let x € X be the unique fixed point of &n°. Its uniqueness implies that & cannot possess more than one fixed point. On the other hand, we have &kn°x = x for any k = 1, 2, . . . whence

\\x-6x\\x = \\&kn°x- &kn°+1x\\x < I

(KTIQ - 1)1

\\x-6x\\x

in accordance with (5.20). Passing to the limit for k —>• oo we come to the equality ir = &x. • Using the inequality (5.15), assumptions (A 6 ), (A 7 ) and the Gronwall inequality we obtain the following estimate for the dependence of a solution upon its initial value x0. If x'(t),x"(t) are solutions of the equation (5.18) corresponding to the initial values x'0, XQ then

E\\x'(t) - x"(tW < eCtV\lx'0 - a#|p.

(5.21)

5.4.2. Markov processes. Suppose that F is a non-random function satisfying (with obvious simplifications) the conditions (Ai)-(A 7 ). Below we assume 7=1. Let xStZ(t) be a solution of the equation

t x(t) =z+ I I F(a,x((r),u)qa(da,du),

(5.22)

s K

zeK, s>0. It can be shown in a standard way that xs
K. Denote by H(K) the set of all real-valued bounded functions on K satisfying the Lipschitz condition

M*') - <(>(z")\ < C\\z' - z"\\. It follows from the estimate (5.21) that the stochastic evolution defined by the equation (5.18) preserves the space H(K): if
\E
s
In our next result we show that the generator of our Markov process acts on functions from H(K) as a hyper-singular integral operator. Here we need instead of (As) a stronger assumption: (A'5) f sup \\F(t,x,u)\\-\\u\ra-1du
Construction of processes

209

Proposition 5.2. If tp € H(K) then

— S

>(* + n*, *, «)) - vWJIIull- 0 - 1 ^,

(5.23)

uniformly with respect to z £ K. Proof. Consider the process

t xs,z(t) -z+ I I F(o-,z,u)qa(da,du) s K

and set y(t) = xs>z(t) — xs
y(t) ~ I I [F(cr,xs^(a),u)-F(ff,z,u)}qa(d(7,du), s K

whence

s K t


t

a-

< C I da 1 1 E\\F(T, XS,Z(T), u)\\-Ka(dr, du) s

s K

t

a


s

K

uniformly with respect to z € K.

Chapter 5

210 Hence the right-hand side of (5.23) can be written as lim ————__———,

t—>s+0

(5.24)

t —S

and the transition to (5.24) is uniform with respect to z. Approximate, for each fixed z,the function F(a, z, u) by step functions Fn(ff,u), n = l,2,... so that Fn ->• F'm sup E

t I I Fn(a,u)qa(da,du)

s
I I F(a,z,u)qa(da,du)

0,

n —> oo.

s K

We may assume that Fn(a, u) = 0 for \\u\\ < q~n, n = 1,2,.... If

t n

X(

\t) = z+ f jFn(a,u)qa(da,du) s K

then

sup E||x(n)(i) - xs,z(t)\\ —» 0,

n -> oo.

(5.25)

s
Fixing n denote by {
ll&(^-)-^(^-0)||>ff-n. Let the sequence {
with probability 1. Here gm = £a(<7m) ~ ^(^m — 0). Since the function a;(n) (i) has almost surely a finite number of jumps on the interval [0, t] being constant between them we can express an increment of (p(x^(t)) as a sum of corresponding jumps. Thus we come to the representation

t = j j [v(x(n}(
- 0))] qa(da,du)

(5.26)

Construction of processes

211

(compare to a similar reasoning in the proof of the Ito formula for real stochastic integrals [58]). The real-valued integrand in (5.26) is bounded, ^.-measurable and vanishes for \\u\\ < q~n. We can insert expectation inside the integral in (5.26) which results in

t

= j JE \>p(x^(a - 0) + Fn(a,u)) -
Since x^ is continuous in the mean, and (p is Lipschitz we find that E
a(da,du).

(5.27)

s K

Our next task is passing to the limit in (5.27) for n -> oo. Note that Fn is constructed from F by fixing its values on sets of small oscillation. This process can be performed in such a way that \\Fn(atu)\\<\\F(a,z,u)\\

(one should choose the values or their left limits at minimum points) . Using the relation (5. 25), the Lipschitz property of (p and the assumption (As) (or (A 5 )) we apply the dominated convergence theorem and obtain the formula

t = j j E [(p(x.,z (a) + F(a, z, u)) -
(5.28)

s K

The internal integral in (5.28) is right continuous as a function of a at the point a = s, uniformly with respect to z, due to the assumptions (A 3 ) and (Ag). Dividing both sides of (5.28) by t - s and passing to the limit for t -> s we obtain the desired equality (5.23). •

212

5.5

Chapter 5

Rotation-invariant processes with independent increments

Let K be a finite extension of Qp. In this section we give a description and explicit construction of all rotation-invariant temporally and spatially homogeneous processes with independent increments on K. Let Y (t) be a temporally and spatially homogeneous stochastically continuous process with independent increments on K, F(0) = 0, p,t is the distribution of the random variable Y(t), ftt(z) is the corresponding characteristic function, that is = / K

Suppose that Y(t) is symmetric, that is all the measures fj,t are invariant with respect to the reflection: Ht(T) = //t(—F) for any Borel set F C K. It is well-known (see e.g. [10, 56]) that

(X(xz) - l)F(dx)

=exp

zeK,

(5.29)

K\{0}

where x is a rank zero additive character, F is a symmetric
Lemma 5.2. The following conditions are equivalent. (i) For each m 6 Z and t > 0

hm,t(x) = Qt(x,Bm(0)) = Qt(0,Bm(-x)) depends only on the absolute value \\x\\. (ii) ||y(£)|| is a Markov process on \\K\\ = {||o:||,3; 6 K}, that is, for

any real function g on K , such that the value g(x) depends only on \\x\\, it follows that whenever \\x\\ = \\x'

Construction of processes

213

(iii) The Levy measure F is such that

(5.30)

F(r) = F(uT) for any Borel set T if \\u\\ = 1. Proof. (i)=>(ii). If g is such as in (ii), then 9 =

k

for some bm, 6_oo 6 M, where wr is the indicator of a set F. Then

x) = y^bm (hmtt(x) - hm-itt(x)) + 6_oo lim hmit(x), *•—J

— —

which depends only on ||x||. (ii)=^(i). Let g = ^Bm(o)- If \\x\\ — \\x'\\, then

hm,t(x) = Eg(Y(t) +x)= Eg(Y(t) + x') = hm,t(x'). (i)=»(iii). Let u £ K, \\u\\ = 1. Then

u-1*) - j x(zx)Qt(udx} = j x(zx)Qt(dx) = Qt(z), whence

exp t

I

(x(xz) - l}F(dx)

=exp t

I (\(xz) - l)F(udx)

K\{0}

K\{0}

for all t, so that F(T) = F(uT) for any Borel set T. (iii)=>(i). Put Q^^r) = Qt(uT), \\u\\ = 1, and let F be the Levy measure of Qt. Then for any z € K

t j

(X(xz)-l)F(u dx)

K\{0}

= exp

/ K\{0}

- l)F(dz) = Qt(z),

214

Chapter 5

whence Q(tu) = Qt. •

Now, if (Yj, Qt) is any rotation-invariant process on K, then by Lemma 5.2, its Levy measure must satisfy the condition (5.30). Set

a(m) = F(K\Bm(Q)). Then the sequence {o(m)}mez is such that a(m + 1) < a(m), m e Z, lim o(m) = 0. m->+oo

(5.31) (5.32)

Conversely, for a given sequence {a(m)}m^z of non-negative numbers with the properties (5.31) and (5.32), there exists a unique measure F on K \ {0} satisfying (5.30) and

F(K\Bm(0))=a(m).

(5.33)

Indeed, every ball V in K, which does not contain 0, lies within a certain sphere SN centered at 0, and any other ball within the same sphere is obtained from F by a rotation. Thus F(V) can be recovered from F(Sff), while the latter is determined by (5.33). In other words, a sequence {a(m)}TO6z satisfying (5.31) and (5.32) corresponds in the one-two-one way to a rotation-invariant process (Yt,Qt), whose Levy measure is given by (5.33). In fact the transition probability can be written explicitly. Denote Pm(t) = (1 - q-1) i=0

t > 0, and let

Pt(x, Bm(y}) = Pm(t), if \\x - 2/|| <
< 1 ^/^ —— — (n /7 1 ~ t ^/ /'P , ^(.^ ^/"i — JPy^-j-jy^l , 1 ^fc^ ^^j j ^t/ — —— 1 l y i~ (/ J^jTj-f-jy

±t\JiiIJrri\y})

if

^ ^4^1

^O.O^/

||ar-0||=>l.

Theorem 5.4. PI defines the transition probability of a rotation-invariant non-exploding process Xt, whose Levy measure satisfies (5.33). Proof. It follows from (5.31) that Pt(x,Bm(y)) > 0. Since every open subset F of K is a disjoint union of balls, we can define Pt(x, F) in an obvious

Construction of processes

215

way obtaining a Borel measure on K. By (5.32) Pm(t) —>• 1 as m —> oo, so that Pt(x,K) = 1. It is clear that Pm(0) = 1, whence P0(a;, .K" \ {a:}) = 0. Let us consider the family of measures $t(F) denned initially on balls as = P m («), if

\\y\\
(Pro+^) ~ P™+»-i W),

if

|M| =
+I

(5-35)

>>l.

It is clear that Pt (x, F) = $ 4 (F—z) (of course, this is just a relation between the transition probability of a process with independent increments, and the associated semigroup of measures; see e.g. [43]). The semigroup property of Pt is equivalent to the fact that <J>4 is a semigroup of measures. The latter is equivalent to the property C^ = ?^v

(5.36)

of the characteristic function

®t(z) = I x(xz}$t(dx). K

Below we shall find $4 explicitly, so that the identity (5.36) will be evident. However we have first to study the family of measures (5.35).

Lemma 5.3. //

lira o(m) = oo, then for each t > 0 the measure $4 is

m—> — oo

absolutely continuous with respect to the Haar measure. If lim o(m) < m—>— oo oo, then for each t > 0 the measure $t is absolutely continuous with respect to the Haar measure everywhere except the origin. In both cases its density outside the origin is given by Vt(y) = (q- I)-1?1"™ (Pm(t) - Pm-i(t))

if \\y\\ = qm.

Proof. Suppose that a(m) —> oo as m —> — oo. Then Pm(t) —> 0 as

m -> -oo, for any t > 0. If \\z\\ < qm, then

Bm(z)

Bm(0)

m+v

If \\z\\ = q

, v > 1, then \\y - z\\ < qm implies \\y\\ = qm+v. Therefore

Bm(z)

(5.37)

216

Chapter 5

and this is what was required. Suppose now that a(m) -> W < oo as m -> -oo. In this case

*t({0})=

lira Pm(*) = e-™ > 0,

771—> —OO

so that t is not absolutely continuous at the origin. At the same time, (5.37) holds in this case too. •

Lemma 5.4. The characteristic function $^ is given by

$>t(z} - exp [-(q - l)~l(qa(-n) - a(-n + l))t] ,

(5.38)

where n = logq \\z\\. Proof. If ||z 11 = qn, then

I"

=

/

oo

_

x(xz)$t(dx) + ^

dx

/

-

IWI=5"

By (1.27) we find that

x(xz] dx = < ||*||=
so that

if m > -n + 1,

0,

*•

^_^ (5/~"\ —p 2

/>\ _ („ _ i1\ — 1 / p

^iV' / — •" — «W

W — -/

, /^l _ p

/V\1

V-" — n+1 W — *—n\>')) i

and after simple transformations we come to (5.38). •

Now we can go on with the proof of Theorem 5.4- The expression (5.38) immediately implies (5.36), so that $t is a semigroup of measures, while Pt is a transition probability. It remains to prove the equality (5.33) for the corresponding Levy measure. Let F be the measure defined by (5.30) and (5.33). In view of Lemma 5.4 and the uniqueness of the Levy measure for symmetric processes, it is sufficient to show that j(x(xz) - l)F(dx) = -(q - \)-l(qa(-n) - a(-n + 1)) K

(5.39)

Construction of processes

217

for | z\\ = qn. Since F is rotation-invariant, we may assume that z = (3~n. The integral on the left in (5.39) is actually taken over those x, for which ||a;|| > q~n+1. If ||a;|| = qm, m > —n + 1, we write a canonical representation

so that

/ The sphere Sm = {x 6 K : \\x\\ = qm} can be decomposed into the union of (q — l)qm+n~1 "closed" balls of the radius q~n. By the rotation invariance F(dy) = (q- l)~lq —— n+lF(S

On the other hand,

X(fi~ny)dy — 1, if m = —n + 1, if m > -n + 1.

218

Chapter 5

As a result we get that j(x(xz) - l)F(dx) = -q(q

- l)-l[a(-n) - a(-n + 1)]

K

ro=-n+2

as desired. •

Example. For the process £a (t) considered above, we have

F(dx)

= ira(dx) =

1

f ^ M~a-ldx,

so that 11*11—— 'cfa =

5.6

i:i£lr,—— .

(5.40)

The generator and its spectrum

Let Tt be the semigroup of operators on L%(K) determined by the transition probability Pt(x, F). It can be verified by a direct calculation that

g(x)dx j Pt(x,dy)f(y) K

K

= j f(y}dy j g(x)Pt(y,dx) K

(5.41)

K

for any /,g € Lz(K) (note that it is sufficient to prove (5.41) for the case when / and g are indicators of balls). It follows from (5.41) that the operators Tt are self-adjoint. The semigroup Tt is strongly continuous. Indeed, let / € L2(K) be of n

the form / = ^ CjWj, where Wi is the indicator of a ball Vt of a radius qm,

and the balls are disjoint. Using (5.41) and the semigroup property, we find that n

n

\\Ttf ~ f\\l2(K) = E C^ llmp^(Vi, Vj) - 1qmPi(Vi, Vj)} i,j=l

where

Pt(Vi, Vj) = q~m j Pt(x, Vj) dx = Pt(Vj,Vi).

»=1

Construction of processes

219

As t ->• 0,

E

= 0.

Together with the contractivity of Tt, this implies Ttf —> f for any / 6 L2(K). Now Tt = e~Ht, t > 0, where H is a self-adjoint non-negative operator (the generator of Tt). By the definition,

<J.O

-i f(x)-

f(y)Pt(x,dy) K

whenever the strong limit exists. In particular, the domain 3)(#) contains

V(K). Indeed, T>(K) is spanned by indicators of all balls in K. Hence it is sufficient to consider the indicator function wz,m of a ball Bm(z), m 6 Z, z € K. We have for x e Bm(z) (Huz>m) (x) = lirnr1 [1 - Pt(x,Bm(z))} = ^ q~~*[a(m + i) - q~la(m + i + 1)] = o(m). i=0 m+v

If dist(z, Bm(z)) = q

, v = 1,2,..., then

(- oo

(Huz>m) (x) = q~" I ^ q-l[a(m + v + i) - q-la(m + v + i - 1)] -q(q-

l

v - ) + (q - l)~ a(m + v) \

- l)~l[a(m + v - 1) - a(m + v}}. It is easy to see that the above limits are the limits in the sense of L-z(K). Thus we have found that

-g-"+1(g-a(m + i/)],

if ||a; - z\\ = qm+v. (5.42)

Chapter 5

220

Due to the linearity the formula (5.42) determines H on T>(K), so that T>(K) C Theorem 5.5. The operator H is essentially self-adjoint on T>(K). Proof. Since H is non-negative, it will be sufficient to prove that for some A < 0 the set (H - XI)D(K) is dense in L2(K). Let f-L(H — XI)'D(K), and / ^ 0 on a set of a positive measure. There exist such z G K, m € Z, that

7= y f ( x ) d x ^ Q . Bm(z)

We may assume that / is real, 7 > 0, and 7 > SUP

\ dx

I

Using (5.42) we find that

f/

= a(m)7 + v^ 2J _ T

f(x)(Hwz,m)(x)

dx

»/

.

«/ > a(m)7 - 7

/

f(x) dx

+ z/ - 1) - a(m + v)}

= 7 lim a(m + v} — 0, whence , (H -

> -A (/,

= -A7 > 0

for A < 0, and we have arrived at a contradiction. • Let us study the spectrum of H. It is clear from (5.42), (5.31), and (5.32), that the indicator function of a ball Bm(z) is an eigenfunction of H if and only if a(m) = 0, in which case the eigenvalue is zero. However we shall see that certain linear combinations of the indicator functions give all the eigenfunctions of H.

Construction of processes

221

For any element z € K with the canonical decomposition z = XXj/^f C; G S, and for any m £ Z we can write

o. Then the ball Bm(z) is determined by the elements C = m - f ) . « • * C-m-i. we shall use the notation {£_„,_„,... , C-m-i} instead of Bm(z). oo

Theorem 5.6. For any m e Z and z —

2

C?/^ € K, there is an

}=-m-v

eigenvalue Xm of the operator H given by \m = a(m) + (q- l)- 1 (a(m) - a(m + 1)), and a (q — l)-dimensional eigenspace spanned by vectors of the form ez,m=

X!

Vm-i^C-™-.,,...^-™-.}.

(5'43)

C_m_l€S

where

^2

bf_m_l =0, andu>^_ ^_

^ _ m _ t } is the indicator function

C-™-i€S

of iAe ball {C-m-v,..., C-m-i}The linear hull spanned by the. functions f-t,m, m t Z, z € =R~, j.s dense

in Li(K). Proof, Let us consider the function (5,43), It is evident that supp ez<m C {C-m-v, • • • , C-m-2}- Note also that the assumption ^ &C- m -i = 0 implies the inclusion suppffe S j m C {C-m-i/!--- >C-m—2}-

Indeed, if x £ { f _ m _ v , . . . , (,"_m_2}, then dist(x, {C-m- v , • • - , C-m-2, C-m-l}) = g'" + "+1,

where n > 1 does not depend on C _ m _ i . Now by (5.42) (He,.>m)(x) = -q~n+1(q - l)-1^^ + n) - a(m + n + 1)]$^^__ t - 0. This observation enables us to reduce the condition Hez^m — he,^m, /?. G Cs to the system of algebraic equations

,C&c = 0, 6C = 0,

??t5,

(5.44) (5.45)

222

Chapter 5

where a^ = a(m) — h, £ G 5, anf = -(q-l)-1[a(m)-a(m+l)],

77 ^ C-

By elementary algebra (for the necessary determinant see [98]) we show that the system (5.44)-(5.45) has a non-trivial solution if and only if h = \m, and in this case the system is equivalent to the single equation (5.45). Therefore we obtain a (q — l)-dimensional eigenspace corresponding to the eigenvalue A m . In order to prove the second assertion, it is sufficient to show that for any z € K and m £ Z, the indicator function u>z<m can be approximated by linear combinations of the eigenfunctions ez
We know that the condition (5.45) determines q - 1 linearly independent vectors e$ n (k — 0, 1, . . . , q — 2), which can be obtained by setting £

0,n=

S

^--i^C-n-!}'

C-n-l€S

where the coefficients are defined as follows. Let us enumerate 5 identifying the values of C-n-i with £ = 0, 1, . . . , q - 1, in such a way that the element from P is identified with ( = 0, and set

With this choice of &£ (5.45) is satisfied for each k, and the functions BQ n are pairwise orthogonal in L^K). We also have + 2).

The theorem will be proved if we show that for any m 6 Z oo

q— 2

n=m k=0

Construction of processes

223

For n > m we have (a) 0 ,m,e§ >n ) i2(/f) = qm since Bm(Q) n {C-n+i} = 0

for C-n-i ^ -P, and 6g = 1 f°r all &• Hence oo g-2 I eO,n\\L2(K}

I] 5] n=m k=0 oo

g-2 i=0

We have used that ^2 k,k+1\ = j^y, which can be easily verified by induction. This completes the proof. • It follows from Theorem 5.6 that the operator H has a pure point spectrum of infinite multiplicity. Note that an explicit construction of a complete orthogonal system of eigenfunctions is known at present (see Chapter 3) only for the operator Da, which (as we saw above) corresponds to the case

5.7

Recurrence and hitting probabilities

In this section we continue the investigation of the rotation-invariant processes (Xt,Pt) described in Theorem 5.4. We shall give criteria for such a process to be recurrent or to hit a single point with a positive probability. Theorem 5.7. The process Xt is recurrent if and only if

£^y = 00-

(5 46)

-

Proof. For m € Z f

°° / Pt(x, Bm(x)) dt = q~l(q - I) 2 ^ q-l[qa(m + i) -a(m + i + I)]"1 i=0 o 00

_

m—I/

_ i \ 2 V~*

— ir

/-\ _

I • , 1\1 — 1

224

Chapter 5

which diverges for any m if and only if oo

-^0^-0(1 + I ) ] - 1 =00. Since (q — l)a(i) < qa(i) — a(i + 1) < qa(i), we come to (5.46). • Denote rx = inf{£ > 0 : Xt = a;}, x

x 6 K.

x

Let C = C {x} (A > 0) be the A-capacity of a one-point set. General properties of these objects are summarized in the following well-known proposition, which holds for any spatially homogeneous standard Markov process on K.

Proposition 5.3. (i) Suppose that for any t > 0 Pt(x,dy) has a density Pt(x,y) with respect to the Haar measure. Then Cx > 0 if and only if the function

gx(x) =

exp(-\t)pt(Q,x) dt

is bounded. (ii) If for some A > 0 Cx = 0, then Cx = 0 for all A > 0 and P(TX < oo) = 0 for almost all x.

(iii) If Pt has a density and if gx(x) is bounded and continuous, then E[exp(-Arx) : rx < oo] =

for any x £ K.

For the proof see e.g. [116]. • Applying Proposition 5.3 to our process (Xt,Pt) we obtain the following criterion regarding A-capacities.

Theorem 5.8. Suppose that a(m) -» oo as m -> -oo. Then Cx > 0 if and only if v(5-47) '

Construction of processes

225

Proof. By Proposition 5.3, (7* > 0 if and only if gx(x) is bounded. If ||a;|| = qm, then

A 4- (q — l)~ 1 (ga(m — 1) — a(m))' Since the right-hand side tends to zero as m -» +00, gx(x) is bounded for large \\x\\. Therefore for —oo. It remains to note that (q - l)a(i) < qa(i) - a(i + 1) < qa(i), and we come to the condition (5.47). • Let us consider the probabilities of hitting single points. Theorem 5.9. (i) //

lim a(m) = W < oo, then

m—>—oo

P(rx < oo) = 0, P(r0 < oo) = 1. 0

x ^ 0,

_j

(ii) // lim a(m) = oo and V^ -———r = oo, then m->-oo ^-^ 1 + an) v i= —oo ' P(rx < oo) = 0 for any x € K. o

_i ———^ < oo, then

P(rx < oo) = lim A4.0 /or any x & K .

Proof, (i) Let W^ be the number of jumps of Xt on the time interval (0, i] with the absolute values exceeding 9™. As we know (see [56]; in

226

Chapter 5

fact we have used similar results for the process £ a ), Wt

has a Poisson

distribution with the mean a(m)t. If 6 = inf{t > 0 : Xt ^ 0}, then

P(# > t) = P(no jumps occur on (0, t]) -

lim P(W((m) = 0) = exp(-W<). >OO

Therefore P(0 > 0) = limexp(-Wi) = 1,

so that P(r0 < oo) = 1. Now let x ^ 0. Denote Jt = Xt — Xt-o. For each £ > 0, || Js|| > 0 for only a finite number of values of s € (0, t}, since E(card{s < t : \\JS\\ > 0}) =

lim a(m)t < oo.

ra-»-co

Let ifc be the time of the fc-th jump. We can write

whence oo

P(rx < oo) < ^P(J t l + - - - + Jtk=x).

(5.49)

On the other hand, P(Jti 6 r) =

F(K\{0})'

and F|x\{o} has no atoms (due to the rotation-invariance), so are the distributions of Jtt. Since Jtt are independent, P(Jtl + • • • + Jtk = a:) = 0 for each k. Together with (5.49) this shows that P(rx < oo) = 0. (ii) By Proposition 5.3 and Theorem 5.8 we have P(rx < oo) = 0

(5.50)

for almost all x. On the other hand, for each m € Z the sphere 5TO = {x £ K : \\x\\ = qm] has a positive Haar measure, and P(TX < oo) is the same for all x 6 Sm, since Xt is rotation-invariant. Hence (5.50) holds for all x ^ 0. As we know, in the present case Pt(x,dy) is absolutely continuous,

Construction of processes

227

and we can see (just as for processes on E"; see [67]) that (5.50) holds for x = 0 too. (iii) By our assumption, gx is bounded, and we can easily see that it is continuous. By Proposition 5.3 E[exp(-Ar x ): rx < oo] =

.

Letting A —> 0 gives our assertion. • Example. Consider the process £a(i), a > 0. As we know, in this case a(m) = caq~am, ca = const. We have

lim a(m) — oo.

m—i—oo

The process is recurrent if and only if a > 1. The capacities <7A are positive if and only if a > 1. If a < 1, then P(TX < oo) = 0. Let us consider the hitting probabilities for the case a > 1. In our situation K

so that

K

If ||a;|| = qm, then

qn \ _|_ nan ' "ll"'ll

' 1

n= —OO

For each A > 0 the function
gx(x) x

' -> 1 as A -> 0, for any x & K. Thus if a > 1, then

P(rx < oo) = 1.

228

Chapter 5

5.8

Processes on adeles

In this section we show how to glue processes denned on Qp for each prime number p into a process on the adele ring A. Suppose that we have, for each p (including p = oo, for which we identify Qoo with R, and ZOQ with [—1,1]) a non-negative self-adjoint operator Hp on 1/2 (QP), and the heat kernel hp, the integral kernel of e~tHp satisfying the properties:

hp(t,x,y)>0,

hp(t,x,y)dy = l.

(5.51)

Suppose further that we are given a basepoint b = (600, &2, • • • , bp,...) € A, and there is a finite set S0 of primes such that

JJ / hp(t, x, bp) dx>0

for all t > 0.

(5.52)

Let AI be the set of those subsets of A, which are finite unions of sets of the form U-jf^, where S is a finite set of primes, X is a measurable subset of n Qp.o-'e {0,+,-},and

Here G^ = 0 ^p> while G^ is a restricted direct product of Qp with respect to Zp for p ^ E (that is U£0 is the set of all adeles whose system of S-coordinates belongs to X). Note that A = U® 0 £ AI. The sets U£+ form the basis of the topology in A, and also generate the Borel cr-algebra

B(A). We check that AI is closed under finite intersections and complements, so that AI is an algebra. For complements, we have (TjX

\c _ jj(Xc) -

IrjX

\c _rrX ~ '

n jr(Xc)

IjrX

\c _jjX —

, , rj(Xc)

Considering intersections we observe first that if R is a finite set of primes, R n £ = 0, then IT*

u

_

£,0 —

_ -

TrXx(QR\1R) U

XUR,+

Construction of processes

229

where we used the notation

Qfi = n Q»" z« = n ZP-

Hence it is sufficient to consider intersections of the form U^ We have

ai

n

u+ n u+ _ n u_ = t / , ug,-nu%i0 = u$t1Y,

t/£+ n t£0 u*t+ n [/!"__- 0.

Let ,4n be the set of subsets of An , which are finite unions of products of sets from A\. It is easy to check that An is an algebra. Let T — {ii, . . . ,tn} C E, 0 < ti < . . . < tn. We define a non-negative function p,T on An, initially on sets of the form

and then extend it by finite additivity. Namely, let fj,r(A) = I^XTi}J^T, where = / •" / h"£(tn-

with

t,x,y) = ][[ hp(t,xp,yp), = (& P )p€Si and JS,T is defined by induction on the number of those £ r, for which a(ti) = "—", as follows: (0) If T is empty, then J£T = 1. (1) If a(ti) = "+" for all U € T, then = H / • ' • / ^p(*n -

230

Chapter 5

(2) If a(ti) £ "-" for all tt € T, then J^T = Jg'T,, where T' = {tt e r : cr(ij) = "+"}, and a' is the restriction of a to T'. (3) Otherwise choose a ti 6 r, for which
and the expressions in the right-hand side are already defined by the induction process. Of course, the steps (2) and (3) are based on the equality in (5.51). Let us extend the measures (j,T to the Borel
Theorem 5.10. The measures HT are finite-dimensional distributions of a Markov process on A. Proof. It is easy to see that the measures in question form a consistent family. It remains to use an appropriate version of the Kolmogorov theorem. Recall that a topological space X is called a topological Radon space [22] if every
An = K x (Q>2 x • • • x Q xZ n factors (p1 is the smallest prime larger than p). An is a topological Radon space as a countable product of complete separable metric spaces. For each n the topology in An coincides with the one induced from A n +j , and An is closed in An+1. These properties imply [22] that A is a topological Radon space, as desired. •

Example. Suppose that for p > p$, where po is some prime, we have Hp = apDa, a > 0, and ap > 0 are certain renormalization constants.

Construction of processes

231

Fixing the base point b = 0, we find that for p> Po

f hp(t,x,0)dx =

f

e-W? d$

n=0

n=0

P < oo,

(5.53)

p then the condition (5.52) is satisfied. Note that (5.53) holds if for large p p

C ~ p(loglogp) s>

(see [118]).

5.9

General stable distributions

5.9.1. Normalized sums. The probability distributions Ga,a on a local field K having the functions ga,a(t) — exp(—a||i||a), a > 0, a > 0, as their Fourier transforms (characteristic functions) are naturally seen as non-Archimedean counterparts of stable distributions. So far it was only a formal resemblance, and it is our task now to interpret them as a part of some class of probability distributions having an intrinsic description. Note that the classical definitions in terms of certain functional equations (as well as the generalizations proposed in [147]) do not make sense in our situation, since the absolute value || • || can equal only an integer power of

qWe shall take a different option - the required class of probability distributions will consist of weak limits of normalized sums o

_

r> — 1 f v" _i_

_t_ v"

\

(K z.A\

where Xi,... , Xn are independent identically distributed random variables with values in K, {k(n)} is an increasing sequence of natural numbers, {Bn} is a sequence of elements from K, \\Bn\\ -> oo.

232

Chapter 5

The limit behaviour of (5.54) depends crucially on the behaviour of the sequence k(n) Pn = T7—TT\> "=1,2,.... k(n + I) Passing to a subsequence we may assume that pn —t X,Q < X <1. It will be shown below that analogs of stable distributions emerge when 0 < A < 1. We shall also show (for K = Qp) that the extreme values A = 0, A = 1 (as in the classical theory for real-valued random variables, where k(ri) = n), and also the case of a bounded sequence {Bn}, lead to degenerate (delta) distributions, cutoffs of the Haar measure etc. 5.9.2. Probability distributions of stable type. It is well known (see [114]) that the Fourier transform of an infinitely divisible probability distribution // on K (as on any locally compact Abelian group) has the form

(x(tx) - l)$(efa),

t € K,

(5.55)

K\{0}

where (l, N{>

if \\t\\
\0, if \\t\\ >qN,

N e Z U {00} (0<x>(i) = 1), x is a rank zero additive character on K, $ is a Borel measure on K \ {0}, which is finite on the complement of any neighbourhood of zero. Formula (5.55) differs from a similar formula for K in two respects - possible presence of the factor fi;v(£) (thus fi may vanish on an open set), and non-uniqueness of the Levy measure $ (this non-uniqueness is typical for locally compact Abelian groups possessing compact subgroups; for example, $ is non-unique for K — Qp, p ^ 2). However, $ can be uniquely determined if the integral under the exp is given. Lemma 5.5. //

tp(t) =

I (x(tx) - I ) $ ( d x ) ,

teQp ,

(5.56)

K\{0}

then for any open compact subset M C K \ {0} = / (y)m(y) dy

(5.57)

Construction of processes

233

where m is an inverse Fourier transform of the indicator function WM of the set M. Proof. We have UM(X) = I X(xy)m(y) dy ,

x € K ,

K

whence m € T>(K) and

/ m(y) dy = WM(O) = 0. K

(5.58)

Using (5.56) and (5.58) we obtain that

If K\{0]

UM(x)$(dx)

=

f/

$(dx) fI (x(xy) - l)m(y)dy = If
K\{0}

K

K

which is equivalent to (5.57). Our use of the Fubini theorem was based on the fact that supp m C {y £ K : \\y\\ < q1} for some / e Z, and X(xy) - 1 = 0 for \\y\\ < ql, \\x\\ < q~l while the measure $ is finite on the set {x e K : \\x\\ > q~1}. •

Let us consider the normalized sums (5.54) with \\Bn\\ -»• oo and pn -t A , 0 < A < 1. Let 7n = -j^*—. Since ||.Bn|| -> oo, there exists a subsequence {7n,} for which ||7n,|| < q~1. We may assume (passing if necessary to a subsequence once more) that 7n —>• 70 in K, ||7o|| < q~l. Suppose that the distributions Fn of the normalized sums Sn converge weakly, Fn —> G, and g(t) is a characteristic function of G. Let f ( t ) be the characteristic function of each of the (independent, identically distributed) random variables Xn. Then / ( -5-

->• 9(t) , n -> oo ,

(5.59)

uniformly on compact subsets of K. The left-hand side of (5.59) will be denoted fn(t).

Proposition 5.4. (i) //A ^ 0 then uniformly on compact subsets of K

l/»(7n*)l ~> lfl(<)l A . If A = 0, i/MS relation holds for those t where g(t) ^ 0.

(5.60)

234

Chapter 5 (ii) The identity \9(jot)\ = \g(t)\^

(5.61)

is valid for any t £ K, if X ^ 0, and for any t with g(t) ^ 0 if X = 0. Proof. Let us consider a random variable Sn ~ Bn+i \X\ + • • • + Xf.(n))

•

Its characteristic function equals /(^))

" =/n(7n<)-

(5-62)

On the other hand, N

f

k(n)

Bn+i

so that (5.59) and (5.62) imply (5.60). Given e > 0, we find, for any fixed t 6 K, such no that )\ < £ if n > no

(since the sequence {7«£}«>o is pre-compact). Thus /n(7«i) -> g(jot) by continuity of 5, and (5.61) follows from (5.60). • Corollary 5.2. // A ^ 0 tfzen p(f) 7^ 0 /or any i 6 K.

Proof. Suppose that g(t0) = 0 for some t0 £ K. By (5.61) we find that \9(to)\ =

Since 7^ —> 0 and g(0) = 1, we obtain that 5(7^^0) 7^ 0 for a certain n, so that we come to a contradiction. • Now we consider the probability distributions on K seen as counterparts of classical stable distributions.

Theorem 5.11. (i) Let $ be a Borel measure on K \ {0}, which is finite outside any neighbourhood of zero and satisfies the relation (5.63)

Construction of processes

235

with 0 < A < 1, 70 £ ./f , 0 ^ ||7o|| < Q~1> for any compact open subset M C K \ {0}. Then a function g(t) of the form

= exp /

(X(ty) - l)$(dj/)

(5.64)

K\{0}

is a characteristic function of a distribution which is a weak limit of some

sequence (5.54) with pn —> A, 7« —>• 70(ii) If the distributions Fn for a sequence (5.54) with independent symmetric identically distributed random variables Xn, \\Bn\\ —> oo, 7n —> 70, 0 ^ \\IQ\\ < q~l, Pn -+ A, 0 < A < 1, converge weakly to a distribution G, then its characteristic function is of the form (5.64) where the Levy measure <J> is symmetric and satisfies (5.63). Proof. (i) By [114], the function (5.64) is a characteristic function corresponding to a random variable X. Let Xi,X^,... be independent copies of X. Set Bn = 7^™, k(n) — [A~™] where [•] means the integer part. Then

/n(0 =

= exp { \n [\-n] 1

L

J

I (x(ty) - l)$(dj/) }• —> g(t) , n ->• oo,

I

..

N

<

'

^

-J-

f

m

—•

uniformly on compact subsets, since [A~n] = A~™ + 0(1), n -» oo. (ii) Let us proceed from the relation (5.59). Since / is real-valued (due to the symmetry of F), continuous, and /(O) — 1, we see that fn(t) > 0 for each fixed t, if n is large enough. Hence, g(t) > 0, and by Corollary 5.2, g(t) > 0 for all t £ K. The sequence {log/n(i)} is bounded, uniformly with respect to t from any compact subset of K. Thus

k(n) ((fn(t))l/k(n} \

- l) = k(n) (exp (J-log/^i)) t \ \K\n) ) K n

( )

so that

*(") ( ( f n ( t } } l / k ( n )

- l) —> log fl («) , n -> oo,

uniformly on compact subsets of Jf.

(5.65)

236

Chapter 5

Introducing the measures

*n(dy) = k(n)F(d(Bny))

(5.66)

where F is the distribution of each random variable Xi, . . . , Xn, . . . , we may rewrite (5.65) as

(5.67) K\{0}

Note that for each t ^ 0 the integral in the left-hand side of (5.67) is actually taken over the set of those y for which \\y\\ > \\t\\~1.

Denote the left-hand side of (5.67) by M, then by Lemma 5.5

= f
(5.68)

K

and by virtue of (5.67)

/

uM(x)$n(dx) —> I m(t)\ogg(t)dt , n -> oo.

K\{0}

(5.69)

K

It follows from (5.69) that the sequence / K\{0}

converges for any locally constant function w with a compact support not containing the origin. Every continuous function with a compact support on K \ {0} can be approximated uniformly by such functions (see [154]). By (5.68), the sequence of measures {$„} is bounded on compact subsets of K\ {0}. This means that {$«} is a Cauchy sequence with respect to the vague topology [56], which is sequentially complete. Thus $n is vaguely convergent to a symmetric Radon measure $ on K \ {0}. Now, in order to prove the representation (5.64), it is sufficient to show that $„ -> <J> in the weak sense on each set Mji00 = [x € K : \\x\\ > q'}, i 6 Z. By Theorem 1.1.9 of [56], that will be proved if we show that $n(-Mj,oo) —> $(Mj)00) < oo , n ->• oo.

(5.70)

Construction of processes

237

Simultaneously (5.70) would imply the required finiteness of $ outside any neighbourhood of the origin. Consider the set M,,i = {x£K: qi+l < \\x\\ < q1} , I > i.

Let us compute 3>n(Miti) using (5.68), where m(t) corresponds to the set Miti. This set is a set-theoretic difference of two balls. We have m(t) = mi(t) —mi(t), where ,, ro (t)

fl,

if ||*|| <«-',

' -\o, if ||*|| >,-',

and we obtain that


(5.71)

whence fn(t)dt

(5.72)

It follows from (5.67) and (5.71) that

\ogg(t)dt. This yields

logg(t)dt. Comparing with (5.72) we come to (5.70).

It remains to prove the relation (5.63). As we have seen, co, uniformly on compact subsets of K. Thus by Proposition 5.4,

Vn(7n*) —> Alogp(t) , n -> oo,

(5.73)

uniformly on compact subsets. Let M be a compact open subset of K \ {0}. Then

M) = lim M-y^M). n—>oo

(5.74)

238

Chapter 5

Indeed, let u>n be an indicator of the set 7~ 1 M,n = 0 , l , 2 , . . . . Writing the action of a measure as a functional we get

{$„ , wn) - {$ , w0) = {$„-*, wn) + {$ , wn - u>0).

(5.75)

For large n 7r^~1M C MIIJ.H where /', /" are certain fixed numbers. As above, this means that the supports of the inverse Fourier transforms wn of all <jjn lie in a certain compact set N, so that

|<*n - $ , w n )| < / \ 0 J

N

due to the uniform convergence. The second summand on the right in (5.75) tends to zero due to the dominated convergence theorem, so (5.74) has been proved. Next, by (5.68)

) = [
X(~ty)dy = \'Yn\-1m('Y

whence K

Now it follows from (5.73), (5.74) and (5.69) that o^M) = \jm(t)\ogg(t)dt

=

K

which is equivalent to (5.63). •

Example. If ||7o|| = q~~l then the relation (5.63) means that $ is determined by its restriction to the group of units U = {x 6 K : \\x\\ = 1}. For a particular example, let the above restriction be proportional to the restriction of the Haar measure:

Qa — 1 f $(M0) = a _ _a_1 dx , a > 0, a > 0, MO

Construction of processes

239

for any open and closed subset M0 C U. Suppose that A = q~a, k(ri) = [qan], 70 = 0. If M is a compact open subset of K \ {0} then it may be written as a finite union

M = \J(MnSN),

SN = {x£K: 11x1) = ^}.

N

In accordance with (5.63),

$(M) = ^ $(M n SN) = ^ q~aN3>(/3NM n U) r

- 2^,1 N

,

J

I

f x

— 2^1 *

J

I

2/>

so that

M Now the identity

(Proposition 2.3) shows that in this case the limit characteristic function g(t) coincides with the function ga,a(t} mentioned in the beginning of this section. Let us consider conditions for the weak convergence of the sequence (5.54) with ||Bn|| ->• oo. Theorem 5.12. In order that the sequence (5.54) be weakly convergent, it

is sufficient that the measures (5.66) converge weakly on each set M,i00 , i € Z, to a measure $ on K \ {0}, finite outside any neighbourhood of zero. If the random variables X±, X%,... are symmetric and 0^0, this condition is also necessary. Proof. The necessity was proved in the course of proving Theorem 5.11. To prove the sufficiency, write /„(£) in the form *(n)

K\{0}

240

Chapter 5

For every fixed t we have

(X(ty) - l)«Md») — >

(x(ty) -

since x(fy)-l = 0 when llyll < lltll- 1 . Recalling that

1 + ——

->•

uniformly on compact sets, we find that

fn(t) —> exp I

/

(x(ty) — l)$(dj/)

, n ->• oo.

(5.76)

Since the function in the right-hand side of (5.76) is continuous, Theorem 3.3.1 of [49] implies weak convergence of Sn. • 5.9.3. Degenerate cases. Considering degenerate cases we assume that K — Qp. We begin with the following lemma. Lemma 5.6. Let n be a probability measure on the Borel a-algebra ofQp.

If \fl(to)\ = I for some t0 e Qp, t0 ^ 0, then f, is a locally constant function. If\fl(t)\ takes only two values, 0 and 1, then there exists £ € Qp, N € Z such that

fl(t) = Xp(t^)^N(t)

(5-77)

where ^N
0, if\t\\Pp>PN. In this case n(dx) = pNft-N(x-£)dx.

If\fl(t)\

= 1 then fi(t) = xP(t&, V =

Proof. Let t0 £ Qp be such that t0 ^ 0, |/*(
the set of rational numbers of the form p~n (OQ + a±p + • • • + an-ipn~l),

n> 1; a 0 ) . . . ,a n _i £ {0,1,... ,p-l},a0 ^ 0. Suppose that //(io) = e 27rir , 0 < r < 1. Then by the definition of XP

e2"ir = I e\p(2m{t0x}p)fj,(dx) whence

r

= 0.

Construction of processes In particular,

241

r

(1 - cos2w({t0x}p - r)}n(dx) = 0.

This means that either r £ Rp or r = 0. In both cases there exists £ e Qp such that r = {£o£}P »that is fi(to) = Xp(to£)- As above, we obtain that

(1 - cos27r({t0(z - £)}P)v(dx) = 0, so that /j, is concentrated on the set of those x for which {t0(x - £)}p = 0, that is on the set {a; € Qp : x — £|p < \to\p1}Now

y xr(*aoM A* = <5g. Otherwise we come to (5.77). The expression for // follows from the integration formula (1.26). • Let us consider the "extreme" cases A = 0 and A = 1 in the weakly convergent scheme (5.54).

Proposition 5.5. //A = 1 then G is degenerate. If A = 0 then either G is degenerate or its characteristic function g has a compact support. If A = 0 and \jo\p = p~l then g(t) coincides with the right-hand side of (5.77). Proof. Let A = 1. As before, we may assume that -yn —> 70, |7o|p < p"1. By Proposition 1, we have

)l = \g(t)\ for any t 6 Qp, so that

Since 7^ —> 0 for n —>• oo, g is continuous and <;(0) = 1, this implies the identity |
242

Chapter 5

has a compact support in accordance with (5.55). The last assertion of the proposition follows from Lemma 5.6. • Finally, let us consider the case of a weakly convergent sequence (5.54) with \Bn\p -ft oo. Then there exists a subsequence Bni -> b € Qp.

Proposition 5.6. If b ^ Q then g coincides with the right-hand side of (5.77). Ifb — 0 then G is degenerate. Proof. Let b ^ 0. It follows from (5.59) that the inequality \ f ( b ~ l t ) \ < I implies the equality g(t) — 0. If |/(6~4)| = 1 for some t then Lemma 5.6 shows that / is locally constant. Thus |/(s)| = 1 when s belongs to a certain neighbourhood of the point b~1t; in particular, \ f ( B ~ 1 t ) \ = 1 for large /, whence \g(t)\ = 1. It remains to use Lemma 5.6. Let b — 0. Take, for an arbitrary s 6 0, we find such a natural number /0 that for I > lo n,

- \g(t)\ < e

, t 6 C.

In particular, for t = Bn, s we obtain that

\\f(s)\n'-\g(Bnis\\<e,

1>10.

Since g(Bnis) -» 1, we see that |/(s)|"< -)• 1 for / ->• oo, whence |/(s)| = 1. By (5.59) G is degenerate. •

5.10

Comments

The theory of stochastic integrals and stochastic differential equations expounded in Sect. 5.2-5.4 was developed by the author [84]. The description of all rotation-invariant processes with independent increments on a local field (Theorem 5.4) was found by Yasuda [158], who also proved their properties given in Theorems 5.7-5.9 and found for this general situation a representation similar to our Theorem 5.2. In fact the class of processes described in Theorem 5.4 was first constructed for K = Qp by Albeverio and Karwowski [3, 4], whose idea was to start from a Markov process on a family of balls covering Qp, defined in such a way that the limit, as balls shrink to points, gives a process on Qp. They found the expression (5.34) for the transition probabilities. Later Yasuda [158] showed that any rotation-invariant process can be obtained by means of this construction. Theorems 5.5 and 5.6 about generators and their spectra are taken from [4]. For various generalizations see [5, 65, 110].

Construction of processes

243

Processes on adeles were studied by Blair [16], and Karwowski and

Vilela Mendes [65]. In this connection we mention also some adelic models in mathematical physics; see [107, 23]. General properties of Levy processes on a wide class of totally disconnected Abelian locally compact groups were investigated by Evans [32]. A description of stable-like probability distributions on a local field is taken from [88]. So far a complete description is known only for the case of symmetric distributions. This can be connected with the fact of nonuniqueness of the Levy-Khinchin representation of an infinitely divisible distribution, a property, which is quite different from the classical case.

Chapter 6

Analysis over Infinite Extensions of a Local Field 6.1

Measurable vector spaces

6.1.1. Definitions. Let E be a vector space over a local field K, B a cr-algebra of subsets of E. The pair (E, B) is called a measurable vector space if the mapping

(x,y)*->x + y,

x,y 6 E,

from E x E to E, and the mapping (A,x) i-> \x,

A e K, x € £,

from K x E to E are measurable with respect to the
Proof. Let us show that the map ( x , y ) t-> x + y is measurable. Consider the cr-algebra

0 = { C 7 € B : { ( x , y ) : x + y <= C} £ B x B}. 245

246

Chapter 6

Suppose that Ti,... , Tn € F and that BI ,... ,Bn are Borel subsets of K. Then n

{ ( x , y ) € T1-l(B1)n...nT-1(Bn)} = f|{(*,») : IX*) + Tt(y) 6 5,}. n

As the map (x, y) *-¥ Ti(x) + Ti(y) is measurable, we see that f| Tt (Bi) 6 t=i ©. On the other hand, such sets generate the cr-algebra B. Therefore its sub-cr-algebra & coincides with B, as required. The proof that the map (A, x) H> Aa; is measurable can be carried out in a similar way. •

Note that (E,E*,B(E)) is a coordinated triple for any separable Banach space E over K. Just as in the case of Banach spaces over E [148], this is a consequence of the non-Archimedean version of the Hahn-Banach theorem (see [134, 127]). In Sect. 6.3 we shall study another example of a coordinated triple. 6.1.2. Gaussian random variables. Before introducing Gaussian measures over a measurable vector space we recall the definition of nonArchimedean orthogonality.

Definition 6.1. Suppose that (Y, \\ • ||y) is a normed space over a local field K. We say that a set X C Y is orthogonal if for any finite subset

{xi,... ,xn} C X and each \i,... ,Xn € K we have = i m^C n l|A i ||-||x i ||y. Y

An orthogonal set X is called orthonormal if \\x\\y = 1 for any x € X. In our next definition Y - K2, ||(Ai, A 2 )||y = max(||Ai||, ||A2||).

Definition 6.2. Let (E,B) be a measurable vector space. Suppose that£ is an E-valued random variable. We say that the probability distribution of £ is a K-Gaussian probability measure (or, equivalently, £ is a K-Gaussian random variable) if when £^,£^ are two independent copies of £, and (An,Ai2), (A2i,A22) £ K2 are orthonormal, then (£^,£^ 2 ') has the same law as

Analysis over infinite extensions

247

Of course, this definition is analogous to one of well-known equivalent definitions (see [37]) of a real-valued Gaussian random variable. The class of .?C-Gaussian K-v&lued (scalar!) random variables is not very rich. It contains a random variable, which is almost surely zero, and some others described in the following theorem.

Theorem 6.1. A K-valued random variable £ that is not almost surely zero is K-Gaussian if and only if its distribution is a cutoff of the Haar measure: G dx} = q-n$(q-n\\x\\) dx

(6.1)

for some n 6 Z, or, equivalently, t&K,

(6.2)

where $ is the indicator function of the interval [0, 1], x is a rank zero additive character on K. Proof. Suppose that (6.2) holds for some n € Z. Note that $(max(a, &)) = $(a)$(fc) for any a, b > 0. If £^,£^ are two independent copies of £, and (An, Ai 2 ), (A2i, A22) is a pair of orthonormal vectors

in K2, then

= * («l*lAll + * 2 A 2 l||) $ (<7"|Ml2 + <2A 22 ||)

= $ (qn maxdMn + i2A21||, | =

x

*i

E [x

Prom the Fourier uniqueness we find that

has the same law as (£'1',£^), so that £ is AT-Gaussian. Conversely, suppose that £ is AT-Gaussian. Denote (t). Thus ip(t) £{0,1} for all t £ K. Suppose that to ^ 0 and
248

Chapter 6

in K2 for each A e K with ||A|| < 1, we see that £W + A£<2' has the same law as £ (1) . Thus
Corollary 6.1. If £ is a K-valued K-Gaussian random variable, then £ belongs to the K-Banach space Loo(P),

p) 11*11) • If £ is not almost surely 0, then

dx. Below we shall construct a non-trivial example of a .ftT-Gaussian measure on an infinite-dimensional measurable vector space. Now we proceed in studying properties of /tT-Gaussian measures. We shall start from a zeroone law.

Theorem 6.2. Suppose that (E, B) is a measurable vector space, and M is a measurable subspace of E. If £ is an E-valued K-Gaussian random variable, then P{£ € M} is either 0 or 1. Proof. Let ^ , ^ be two independent copies of £. For n = 1,2,..., set

An = {£(1) + /3"£(2) e M,j3n^ + £(2) g M J

(as before, /3 is a prime element of K). Note that {(l,/3 n ), (/?",!)} is an orthonormal pair. Indeed, it is clear that ||(l,/3")||tf2 = ||(/3n,l)||K2 = 1. It is sufficient to show that there is no z € K such that

||(1, /3") + z(/3n, 1)11*2 = max {||1 + z/3n\\, ||/3" + z\\} < 1. Otherwise we would have ||1 + z(3n\\ < 1, so that ||^/3n|| = 1 and hence \ z\ = qn. Then ||/3n + z\\ = qn > 1, which is a contradiction. By the definition of a JsT-Gaussian random variable we see that

Analysis over infinite extensions

249

has the same law as (£^,£' 2 ^)> and hence

P(An) = P{£ 6 M}P{£ i M}.

(6.3)

If m ^ n, then the matrix

(l

Pn

is invertible. Therefore if both £(1) +/3m^ and £(1) +f3n^ belong to M, them both £(1) and C (2) (and hence both /3m£(:) + £ (2) and j3n^ +, belong to M. Thus Am f\An = 0. It follows that

and from (6.3) we conclude that P{£ 6 M} e {0, 1}. •

Let (E, F, B) be a coordinated triple.

Theorem 6.3. If £ is an E-valued random variable, then £ is K-Gaussian if and only */T(£) is K-Gaussian for all T € spanF. Proof. Suppose that T(£) is /C-Gaussian for all T € spanF. Let ^•^ be two independent copies of £. Just as in the case of real measurable vector spaces [22], we may use characteristic functionals, that is we need to check that

E x T! A n £ 1 + A 1 2 £ 2

+ T2

= E(X(T1(X1) + T,(X2))}.

(6.4)

By Corollary 6.1, the left-hand side of (6.4) is given by $ (max

Suppose that dim span{Ti(£),T2(£)} = 2 (other cases are similar). Let 771,7^2 be an orthonormal basis for span{Ti(£),T2(£)} in Loo(P). We can write +7i2»? 2 , +722?? 2 .

250

Chapter 6

Now

722??2)|U00(P)

= max(||An7ii + A 2 i7 2 i||, ||An7i2 + A 2 i7 22 ||). Similarly,

||(Ai2Ti + A 22 T 2 )£|| Loo( p ) = max(||Ai27n + A2272i||, ||Ai2712 + A22722||). It follows that the left-hand side of (6.4) equals $(max(||7ii(An,A 12 ) +7 2 i(A 2 i , A 22 )||, ||7i2(An, A i2 ) 4- 7 22 (A 2 i, A 22 )||)) = $ (max(max(||7ii||, ||721|| by the orthonormality of (An,Ai 2 ), (A 21 ,A 22 ). For the right-hand side of (6.4) we get $ (max(||7n»?i + 7i27?2||L00(p), H7217/1 + 722%IU Therefore (6.4) holds and £ is JiT-Gaussian. The proof of the converse is straightforward. • Next we consider the behaviour of a Gaussian measure under an additive shift. A full description is given by the following result. Theorem 6.4. Let (E,F,B) be a coordinated triple, and P a K -Gaussian measure on E. Set E = {x € E : \\T(x)\[ < ||T||Loo(P) VT e spanF}.

Let I : E —> E be the identity map. If x E S, then the law of x + I under P isP itself. Otherwise, the law of x + I is perpendicular to P. Proof. If 77 is a jpf-valued, /f-Gaussian random variable, and y € K is such that ||y|| < ll^llz,^, then, since the law of 77 is a Haar measure on the subgroup z : \\z\\ < H^Hi^, we see that the law of y + 77 coincides with the one of n. Therefore if x € S, we have that the law of T o (x + 1) = T(x) + T under P is that of T under P for all T £ span F. Hence the law of x + I under P is that of / under P. If x $. S, then there exists such a functional T € spanF that

Analysis over infinite extensions

251

Then \\T(x + 7)|| = ||T(x)|| P-almost surely, which means that the law of x + I is perpendicular to P. •

Note that in similar results regarding Gaussian measures on vector spaces over R (see e.g. [22] or [95])) the dichotomy of equivalence and perpendicularity is established. The next result shows that in the nonArchimedean case equivalent .ftT-Gaussian measures coincide.

Theorem 6.5. Suppose that (E,F,B) is a coordinated triple. If PI and P2 are equivalent K -Gaussian measures on E, then PI = P2Proof. It follows from our assumption that

for all T e spanF. By Corollary 6.1, for all such T the distribution of T under PI is the same as that under P 2 , whence PI = P2. • 6.1.3. Minlos-M§drecki theorem. A complex-valued function / on a vector space E is called positive definite if /(O) = 1 and

if(xi - Xj)cJ > 0 for all natural numbers n, c; € C, and Xi 6 E. The well-known Minlos theorem (see e.g. [22]) states that any continuous positive definite function on a nuclear locally convex topological vector space E over E is the Fourier transform of a Radon probability measure on the Borel
Theorem 6.6. Let E be a Hausdorff locally convex space over a local field K. Any continuous positive definite function on E is the Fourier transform of a Radon probability measure on B(E*), where the conjugate space E* is equipped with the *-weak topology a(E*,E). Conversely, if E is barreled, then the Fourier transform of a Radon probability measure on B(E*) is a continuous positive definite function on E. In order to prove this theorem, we shall need some auxiliary results. Let us consider the finite-dimensional space Kn with the norm

252

Chapter 6

The set

On = {\=(\1,...,\n)EKn: ||A|U-<1} is a lattice (an open compact O-module) in Kn. The integration formula (1.26) admits a multi-dimensional generalization (see Chapter VII in [156]). In particular, there exists such a lattice O™ in Kn that the indicator function of On is the Fourier transform of the probability Haar measure of the additive group of O". If A : Kn —> Kn is a linear mapping,

= \\AX\\K»,OA = {A € Kn : pA(X) < 1}, and flA(X) is the indicator function of OA, then

PA(\)

(6.5) n

K

where x is a rank zero additive character on K, HA is a probability measure concentrated on a certain O-submodule MA of Kn, XiTJi,

X = ( A i , . . . , A n ), T] = (?/i, ... ,T]n)-

8=1

It is clear that MA is a compact convex absorbing set in Kn, 0 6 MALemma 6.2. \N, K"

where \\A\\pf is the "nuclear norm" of the operator A defined as

\\A\\N = sup \\Aei\\K*

(6.6)

for an arbitrary orthonormal basis {ej}™ of Kn. Proof. Note first of all that ||A||jv does not depend on the choice of an orthonormal basis in (6.6). That follows easily from the fact that linear operators transforming orthonormal bases are isometries of K". It follows from the properties of the set MA mentioned above (see [108]) that there exist a basis {ei,... ,en} of Kn, and positive real numbers o i , . . . ,a n , such that

Analysis over infinite extensions

253

We have by (6.5) that

J x(t(ei,ri))HA(dri)=nA(tei), K"

which, as a function of £ 6 If, equals the indicator function of the set \pA(ei)]-lO. On the other hand, if HA is the probability distribution of the random variable f o , - ) : (MA,HA)-+(K,B(K)), then

I x(t(ei,r,})HA(dr1) = J X(t)HA(dt). Kn

K

Now we find from (1.26) that Oj = pA(ei), so that

) • O.

(6.7)

Prom (6.7) we obtain

= f maxJ(\,ei)\\HA(dX) MA

< max pA(ei) = max ||^iei||/f- = \\A\\N Ki
Ki
as desired. Lemma 6.3. Let n be a probability measure on (K",B(Kn)), fi its characteristic function, and A, C arbitrary linear mappings from Kn to Kn. If

|1 - Re ju(A)| < s for all \€Kn with pA(\) < 1,

then M ({A € Kn :

where pc(\} = \\C\\\K*.

Pc(\)

> 7}) < £ + ^\\AC\\N,

7 > 0,

(6.8)

254

Chapter 6

Proof. It is sufficient to consider the case when 7 = 1. Let He be the probability measure related to the operator C just as HA was related to A. We have

= Re

Under our assumption we get

If J

fI Hc(drj) < s, J PA (»?)
[1 — R-e/*('?)] Hc(drj) <e

pA(n)
[l--Rafi(n)]Hc(dri)<2

f

Hc(drj)<2

f

(6.9)

PA(r,)Hc(drj).

PA(TI)>I

(6.10) By Lemma 6.2

j pA(n)Hc(dn) = j \\Ari\\K~Hc(dri) = j \\n\\KnHAC(dri) < \\AC\\N. K»

K"

Kn

Combining this with (6.9) and (6.10) we obtain (6.8). • Let us introduce some notations. Suppose that V is a locally convex space over K, G C V is an arbitrary subset. The polar set G° of G is defined as

G° = {v* € V : ||(t;, OH < 1 for each v € G} . If p(v) is a non- Archimedean seminorm on V, then we write Np = {v € V : p(v) = 0}.

Lemma 6.4. Let V be a finite- dimensional vector space over K, andp(v) a non- Archimedean seminorm on V . Let fj, be a probability Borel measure on V* such that \l-ReJl(x)\<e i f p ( x ) < l

(e < 1). Then for every 7 > 0

where Bp(j) = {x € V : p(x) < 7}.

Analysis over infinite extensions

255

Proof. Let Wp be a complementing subspace to Np, that is V = NP+WP. Let Tp be a projection in V onto Wp corresponding to this decomposition. It follows from the non-Archimedean (ultrametric) property of p that p(v) = p(Tpv). The seminorm p induces a norm on Wp. Let e i , . . . , e/t (fc = dim Wp) fc be an orthonormal basis in Wp. If v € V, we can write v = VQ + ^3 A»ei, i=l

where WQ £ ./Vp, Aj € /
p(w) = max ||Ai||.

(6.12)

Below we shall use a basis in V obtained by complementing e.\,... , e^ by an arbitrary basis ek+i,... ,e n in Np (if Np ^ {0}). Using this basis we may identify V with Kn. Then (6.12) yields the representation p(\) = \\AX\\K», \&Kn, where A = ( £ $ ) . Now V* is also identified with Kn, with the basis {e,} viewed as selfn

n

dual. If A = £ A i6i £ V, r? = £ r?;e; € V*, then i=l

t=l

By (6.12), Bp(j) is identified with the set of those A, for which ||Aj|| < 7 for i = 1, . . . k. It is clear that the polar set [^(7)]° consists of those r), for which ||T;J|| < 7"1 if i — l,...k, and TH — 0, i — k + 1, . . . ,n. Its complement V* \ [-Bp(7)]° is the set of 77 with either \\rji\\ > 7"1 for some i 6 {1, . . . , k}, or ||77i|| > 0 for some i € {k + I , . . . , n}. Let us consider the sequence of sets MI C V*, I = 1,2,..., consisting of those 77, for which either ||^|| > 7"1 for some i £ {!,..., k}, or INI > T"1?"' for some i 6 {k + 1, . . . ,n}. Obviously M( /• V* \ [Bp(7)]° monotonically, so that

On the other hand,

where C = ( Q a-'j ) , and as usual /? is a prime element of K. Using Lemma 6.3 and observing that 1)^4(711^ = ||A||^ we find that H(MI) < £ + 27, which implies (6.11). •

256

Chapter 6

Proof of Theorem 6.6. Let L C E* be a subspace of a finite codimension (an annulator of a finite-dimensional subspace L° of E). For any subset F C E*/L the set of elements x* € E*, whose images in E*/L under the canonical mapping belong to F, is called a cylindrical set determined by L, with the base F. Cylindrical sets form an algebra Cyl(E*). Its subalgebra Cyl(E*,L) corresponding to any fixed -L is a cr-algebra. Just as in the case of topological vector spaces over R (see [22]), a continuous positive definite function / on E determines a cylindrical measure yit, that is a finitely additive measure on Cyl(E*), whose restriction HL is cr-additive on each Cyl(E*,L), such that

Each HL corresponds to a measure on the finite-dimensional space E* /L. In order to prove that the measure ^ can be extended to a cr-additive measure on B(E*), it is sufficient [22] to find, for every e > 0, such a compact subset Ce (with respect to the topology a(E*,E)) that for any cylindrical set B with B n Ce = 0 we have fJ,(B) < s. Since / is continuous, we can choose from the family of seminorms defining the topology on E such a finite collection {pi,i E I(s)} that |1 - Re/(x)| < s/2

ifpe(x)

rW

= max Pi(x) < 1.

(6.13)

Let Bpl(-y) = {x £ E :

Pe(x)<j},

7>0.

The polar set [Bpc('j)}0 is compact in the topology a(E*,E) on E* (see [108, 145]). This set is our choice for Ce. Let B be a cylindrical set, B n [-8^(7)]° = 0, and L C E* be the subspace of a finite codimension, which determines B. In the quotient space E* / L we have the base set F with the same measure, ^L(F) = ^(5), where we identify a cylindrical set from Cyl(E*,L) with its base F. The image of the polar set B n [Bpc (7)] under the canonical mapping is the polar set of the set Bpe(7)nL° = {x€L°: p£(x) < 7} ,

which lies in the finite-dimensional subspace L° whose dual space is E* /L. As a result we have

(E'/L \ (Bpe (7) n

Analysis over infinite extensions

257

and by Lemma 6.4 <

+ 27 < £

It remains to prove the converse statement. Suppose that fi is a Radon probability measure on B(E*), where E is a barreled space. Let xi , . . . , xn £ E, GI, . . . ,cn € C, EO be the finite-dimensional subspace spanned by xi, . . . ,xn, L its annulator, and /ZQ the measure induced by n in the quotient space E* /L ~ EQ. We have (x) = I X((x,x*))fi0(dx*). Since E£ is finite-dimensional, we may use the analog of Bochner's theorem mentioned in Sect. 1.6 obtaining that > 0.

It is clear that /z(0) = 1. Thus /* is positive definite. By the definition of a Radon measure, for a given e > 0 there exists a *-weakly compact set Cs C E*, such that n(Ce) > 1 - -. Since E is barreled, C£ is contained in a polar set of some neighbourhood of the origin. Thus one can find such a non- Archimedean continuous semi-norm pE on E that for some % > 0. Hence f

(x,x*)\\n(dx*) < e + p'e(x),

where pfe(x)=

sup \\(x,x*)\\ ^*e[B P£ (7,)]° is continuous at the origin, so that Re ft is continuous at the origin. Finally, the positive definiteness of fi(x) implies the inequality |£(s') - $(x")\2 < 2 [1 - Reftx' - x")] ,

x',x" € E,

(see [55], where it was proved for positive definite functions on an arbitrary group). Now it follows that ft is continuous on E. •

258

Chapter 6

6.2

An infinite extension as a topological vector space

6.2.1. Main notions. Let A; be a non-Archimedean local field of zero characteristic (thus k is a finite extension of the field of p-adic numbers). Consider an increasing sequence of its finite extensions

k = K1cK2C...cKnC... . For each n = 1,2, ... we denote by | • |n a normalized absolute value in Kn; if Nn : Kn -> k is a norm mapping then \x „ = \Nn(x)\\. We shall use also a (non-normalized) absolute value ||a;|| = x\n n (x € Kn) which coincides with |or|i for x & k being defined correctly on the infinite extension oo

K — |J Kn. Here mn is the degree of the extension Kn/k. Of course, n=l

the meaning of the notation || • || is different here (and until the end of this chapter) from the one used before. This abuse of notation is hopefully justified by the role of the absolute value || • || over an infinite extension, which is similar to that of a normalized absolute value over a local field. Let us consider, for each n, a mapping Tn : K —> Kn defined as follows. If a; € Kv, v > n, put

where T^Kv/Kn : Kv —> Kn is the trace mapping (note that the degree of the extension Kv/Kn equals m^/mn ). If a; € Kn then, by definition, Tn(x) = x. It follows from the transitivity property of traces that Tn is well-defined and Tn°Tv = Tn for v > n. Below we shall often write T instead of T\ . Note that the equality Tn(x) = 0 for all n implies x = 0. Indeed, if x € K then x € KI for some /, and TI(X) — x. The field K may be considered as a separable topological vector space over k with thejnductive limit topology. Denote by K the projective limit of the sequence {Kn} with respect to oo

the mappings {Tn}. It is defined as a subset of the direct product H Kn n=l

consisting of those x = (xi, . . . ,xn, . . . ), xn 6 Kn, for which xn = Tn(xv) if v > n. The topology in K is introduced by seminorms

Both spaces K and K are complete and reflexive [145, 108].

Analysis over infinite extensions

259

An element x € K may be identified with (x\ , . . . , xn: . . . ) 6 K where xn = Tn(x). Thus K may be viewed as a subset of K. the mappings Tn can be extended to linear continuous mappings from K to Kn by setting Tn(x] = xn for any x = (xi , . . . , xn, . . . ) € Jf.

Proposition 6.1. Tfte topological vector spaces K and K are the strong duals of each other, with the pairing defined by the formula

(x,y)=T(xyn)

where x e Kn C K^_ y = (j/i, . . . , yn, • • • ) € K, yn € Kn. As a subset of K, K is dense in K. Proof. Each field Kn may be identified as a topological vector space over A; with its dual K^ - if tp € K£ then there exists a unique element a-v € Kn such that

(f>(x) - T(avx),

x e #„,

and if may be identified with av. Let z/ > n. The imbedding rnv : Kn -4 ^ determines (due to the above identification) the adjoint mapping T^V : Kv -> /f n . We shall show

that r*^ = Tn\Kv. Indeed, if x E Kn, ip £ K* then

i]>(rnv(x)) =
where