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0 and all a,
H-Q is a bounded linear operator. We will use in the sequel the shorthand notations F e = PeF, A e = PeA, and fe = Pc!. In the following we make a somewhat stronger assumption ensuring global nonlinear stability of our SPDE (2.1). For simplicity, we restrict ourselves to cubic nonlinearities. This assumption is responsible for the global existence of solutions (cf. Proposition 2.1) and for uniform (in t) bounds on IEllu(t)IIP for solutions of (2.1) (cf. Theorem 3.2). Note that these bounds are independent of the initial condition.
45
AMPLITUDE EQUATIONS FOR SPDE
ASSUMPTION 3. Let Assumption 2 be true and assume that the linear
operator A belongs to such that
an: 'It) . Moreover, there exists a constant CA> 0
(Av, v) ::; CA(llvl12
(2.9)
+ (-Lv , v))
for all
vE
u' :
We also assume that:F is trilinear, :F : ('It 1)3 ----+ 'It is continuous and that (2.10) for all vc, Wc E N \ {O} . We finally assume that there exist constants K and,L E [0,1) such that, for some 8 > 0, (:F(v + c/J), v) ::; KIIc/J11 4
(2.11)
-
811vl1 4 -
,L(Lv, v) ,
for any c/J,v E n' , Concerning the stochastic perturbation we will always assume that the following is true. (For a detailed discussion of Q-Wiener processes and stochastic convolutions see [18] .) ASSUMPTION 4. The noise process is formally given by ~ = QOtW , where W is a standard cylindrical Wiener process in 'It with the identity as a covariance operator and Q E £('It, 'It) is symmetric. Furthermore, there exists a constant a < such that
!
(2.12) where" . to 1-£ .
IIHS(1i)
denotes the Hilbert-Schmidt norm of an operator from 'It
REMARK 2.1. Straightforward computations, combined with the properties of analytic semigroups allow to check that Assumption 4 implies the following (see [18, Section 5.4) for the first assertion}: • The stochastic convolution Wdt) = f~ eL(t-s)Q dW(s) is an 1-£valued process with Holder continuous sample paths . • There exist positive constants C and 'Y such that
(2.13) holds for every t > O. 2.2. Note that we do not assume that Q and L commute. Hence , it is in general not true that Q and Pc commute. Therefore , the noise processes PcQW and PsQW will not necessarily be independent, which implies that the amplitude equation (2.4) and equation (2.5) for the second order correction are coupled through the noise. It is straightforward to verify that REMARK
46
DIRK BLOMKER AND MARTIN HAIRER
Therefore, the stochastic convolution is a Wiener process on N and it is a stable Ornstein-Uhlenbeck process on S . This means that the noise acts in two completely different ways on Peu and Piu for some mild solution u. To give a meaning to (2.1) we will always consider mild solutions, which are given by the following proposition. PROPOSITION 2.1. Under Assumption 1, 2, and 4, for any (stocha stic) initial condition Uo E 'H equation (2.1) has a unique local mild solution u. This means we have a stopping time t* > 0 and a stochastic process u such that u : [0, t*] --t 'H is a solution of (2.14) for t
~
u(t) = etLuo
+
it
e(t-T)L[€2Au + F(u)](r)dr
+ €2Wdt)
r.
Suppose additionally that Assumption 3 is true, then the solutions are global, which means t* = 00. For the proof note that the existence and uniqueness of local solutions is standard since we consider locally Lipschitz-continuous nonlinearities. See for example [18, Section 7] we can also apply the deterministic approach of [26, Thm. 3.3.3] path-wise. For LP-theory with application to NavierStokes eq. see for example [10, 11]. The global existence follows from standard a-priori estimates for v = u - WL , as v is a weak solution of the following PDE with random coefficients: (2.15)
The formal idea is to take the scalar product with v, in order to derive estimates for IIvl12 and hence Ilu112. 2.2. Examples of equations. In the literature there are many examples of equations of the type given by Assumptions 1, 2 or 3, and 4. For instance, the well-known Ginzburg-Landau equation (see [20] for a standard proof of existence)
and the Swift-Hohenberg equation OtU = -(~
+ 1)2u + vu -
u 3 + a~ ,
which was first used as a toy model for the convective instability in a Rayleigh-Benard problem (see [27] or [15]), fall into the scope of our work when the parameters v and a are small and of comparable order of magnitude. Both equations are considered on bounded domains with suitable boundary conditions (e.g. periodic, Dirichlet, Neumann, etc.). Other equations could involve nonlinearities of the type o;(u 3 ) or ox((oxu)3). The first nonlinearity is considered with the Sobolev space
AMPLITUDE EQUATIONS FOR SPD E
47
1t = H - 1 , while the second one has to be considered in £2 , provided we have the following Poincar e typ e inequality Il ull :S C lloxull for u E D (L ). Another example arising in t he t heory of surface growth is (2.16)
subject t o periodic bo undary condit ions an d which will ensure a Poincar e ty pe inequality. was rigorously treated in [28]. Here we can 2 2u (J = 0 (E ), whe re /la is such t hat L = - t1
moving fram e fe u dx = 0, The deterministic equat ion consider /l = /la + E2 and /lat1u fulfils Assumption 1.
3. Amplitude equations, main results. We review t he two main approaches to ver ify the approximation via amplitude equa t ions . One relies on a purely local picture and uses Assumption 2, while t he other t akes into acco unt the global nonlinear stability of the equation given by Assumpti on 3. 3.1. Attractivity. The at t ract ivity justifies the ansat z for the form al computation. It shows t ha t afte r a comparably short time the solution is of the form of the an satz (2.3) . THEOREM 3.1 (At t ractivity-local) . Under A ssum pt ion s 1, 2, and 4 there are cons tants e, > 0 and a tim e t e = 0(ln (c 1 ) ) suc h th at fo r all mild soluti ons u of (2.14) we can wri te u( t e ) = m e + E2Re with ae E N and Re E S , where (3.1)
1P' ( llaell:S s, 11 Re11 :S E- K ) 2: 1P'( llua ll :S
C3bE)
- cle-c2e- 2K ,
forallb >O and e c. (0,1). The proof of this result is a st raight forward modi fication of Theorem 3.3 of [4]. It relies on the fact t hat small solut ions of ord er O(e) are on sm all time-scales given by the lineari sed picture, which is dominat ed by the semigroup estimat es (2.7) and (2.8) . THEOREM 3.2 (Attractivity-global) . Let A ssumptions 1, 2, and 4 be satisfied. Th en f or all Ta > 0 an d p 2: 1 there are cons tants c p > 0 such that
(3.2) for all1t- valued m ild solutions u of equation 2.1 independent of the initial conditi on. Furthermore, th ere is a tim e t e = 0(ln(c 1 ) ) such that given a family of posit ive constants {bp} p~l there are positive constants { Cp} p~l , such that for all1t-valu ed mild solutions u of equation (2.1) with IEllu(O)IIP :S bpEP we have
(3.3)
IEllu(t)IIP :S CpEP and IE IIPs u (t + te)IIP :S CpE 2p
f or all t 2: 0 and E E (0,1 ).
48
DIRK BLOMKER AND MARTIN HAIRER
The proof is given by a-priori estimates. This was not directly proved in [7], but under our somewhat stronger assumptions this is similar to Lemma 4.3 of [7] . It relies on a-priori estimates for V o = u - WL - o with = O(€2) , which fulfils a random PDE similar to (2.15). We omit the proof, as it is technical but straightforward.
o
3.2. Approximation. For a solution a of (2.4) and 'l/J of (2.5) we define the approximations €Wk of order k by
The residual of
(3.4)
€W
is given by
Res(€w(t)) = -€w(t)
+
it
+ etL€w(O) + €2Wdt)
e(t- r)L[€3 Aw + F(€w)](r)dr.
In order to show that €W is a good approximation of a solution u of (2.14), the main step is to control the residual. The main idea is to obtain bounds on PeRes(€w) via the amplitude equation and to bound PsRes(€w) by using the stability of the equation which is ensured by our spectral gap (cf. (2.6) or (2.7)). These estimates require good a-priori bounds on the approx imation €Wk, but do not require any a-priori knowledge on the solut ion u of the original equation. Bounds on the residual easily imply approximation results, as we can establish bounds on the difference of euu; and u using (3.4) and (2.14). THEOREM 3 .3 (Approximation-local) . Suppose Assumptions 1, 2, and 4 are true . Fix the time To > 0 and som e sm all r;, E (0,1) . Then there are constants Gatt> 0 and c, > 0 such that for € E (0,1) we obtain for all solutions u of (2.14) and all solutions a of (2.4) (with I; instead of Fe) JP> (
sup tE[O,TQE-2j
lIu(t) - €wl(t)lIx ::::: Gatt€2-K)
2: 1 - JP> (lIuo - m(O)llx 2:
C1 €2-
K) - JP>(lluollx 2:
C2
€) - c3e-e41n(g-1)2 .
The proof of this result is a straightforward modification of Theorems 4.1 and 4.3 of [4] . We use ideas of [9] to allow for weaker bounds on SUPTE[O ,To]la(T)! by c* In(c 1 ) , which were not present in [4] or [5] . There the probability was bounded by terms of order 0(1) in e without any further information on the smallness. Nevertheless, we can easily improve these proofs. In that situation, we can use the following large devi ation bound JP> (
sup TE[O ,To]
la(T)I2: Gln(€-l)) ::::: Ge- cln(g-1)2 ,
AMPLITUDE EQUATIONS FOR SPDE
49
which is not exponentially small, but smaller than any power of c. This relies on the fact that the amplitude equation is nonlinear stable, which follows from Assumption 3. This stability allows to carry over large deviation results for the Brownian motion (3 in N ~ ~n to results for a. Under the stronger Assumption 3 we can prove a much better result. THEOREM 3.4 (Approximation-global) . Let Assumptions 1, 3, and 4 hold and let u be the mild solution of (2.1) with (random) initial value Uo satisfying (3.3). Then for all p > 0, 1 » /'\, > 0 and To > 0 there is a constant Cap p explicitly depending on p and To such that the estimate
holds for e E (0,1) . The proof is Corollary 3.9 of [7] .
4. Applications. We give results for approximate centre manifolds and the dynamics of random fixed points. 4.1. Approximate centre manifold. This section uses the approximation results of the previous section . We rely especially on Theorem 3.4 to extend the results, which were briefly sketched in [4] or [5], by using second order corrections introduced in [7] . That is why we restrict ourselves to nonlinear stable equations given by Assumption 3. Our main result will show th at th e flow along N is well approximated by the solution a of the amplitude equation on a slow time scale. The distance from N is given by a fast oscillation 'IjJ, which is a stationary Ornstein-Uhlenbeck pro cess. And everyt hing is valid only up to errors of order O(c3 - K ) and with high probability. The flow given by (2.1) is approximated with high prob ability as sketched in Figure 1. There the typical behaviour of solutions is given. THEOREM 4 .1. Suppose A ssumptions 1, 3, and 4 are true, then there is a logarithm ic tim e t e = O(ln(c l ) ) such that the following is tru e. For an arbitrary mild solution u(t) of (2.1) with (random) initial condition uo, such that IElluollP ::; JpcP for som e fixed family of constants {<5p}p~l, we denote bya(t) the solution of (2.4) with a(t e ) = Cl Pcu(t e ) . Furthermore let 'IjJ*(t) be the stationary Ornstein-Uhlenbeck process solving (2.5) given by (4.1) . Finally, fix the time To > 0, some small /'\, E (0,1), and any p > O. Then there exists a constant C > 0 such that
holds for e E (0,1) .
50
DIRK BLOMKER AND MARTIN HAIRER
N in X £a(O)
e: 2 'l/J (0)
~ U (O)
( J
2
a ( E: t
g2'l/J(t)
u(t )
FIG . 1. Typical trajectory on the approximate centre manifold .
Proof. First we use global nonlinear attractivity in logarithmic time t~l) for arbitrary initial conditions (cf. Theorem 3.2). Then we approximate with solution aCt) of (2.4) and ;Pet) of (2.5) for times t E [t~l), Toc 2 J. Define a version of the stationary Ornstein-Uhlenbeck process by (4.1)
'l/J*(t)
=
{too e-L(t-s) dPsQW(s) ,
where W(s) = W(s) for s > 0 and it is an independent Wiener process for s < O. For (3 in the amplitude equation, we need only th e rescaling (3(T) = gPc QW (T c 2 ) . The difference between ;p and 'l/J* is trivially small in any p-th moment, if we wait another sufficiently large logarithmic time tF). Define now tE: := t~l)
+ tF).
The difference between aCt) and aCt) is small by the approximation result, because first lIa(tE:) - a(to)11 = O(g3-K) by Theorem 3.4. Then, by the same theorem Ila(t) - a(t)11 ~ lIa(t) - Pcu(t)11 + IlPcu(t) - a(t)11 = O(g3-K). 0 4.2. Dynamics of the random attractor. We can determine the dynamics of random fixed points by the approximation result over a very long time-scale with high probability. It suffices indeed to apply the results of the previous section by starting the equation in the random fixed point.
AMPLITUDE EQUATIONS FOR SPDE
51
Let us first fix some notation. If we consider a two sided Wiener process W = {W(t)}tEIR, then it is well known that solutions of (2.14) define a random dynamical system (e.g. via transformation to (2.15)) . Here cp(t,Uo, W) is the solution u(t) given initial condition Uo and twosided noise path W. A random fixed point ao(W) is a random variable such that cp(t,ao(W) , W) = ao('19 tW) , where t9 t W (s ) = W(t + s) - W(t) . For a detailed discussion of random dynamical systems see [1] . For the existence of random (set) attractors see for example [14, 41, 39]. COROLLARY 4.1. Under the assumptions of Theorem 4.1 let a., be a random fixed point of the random dynamical system generated by (2.14}. Then
where a(O) = Pea o , and a is a solution of (2.4). The proof is basically just a simpler case of Theorem 4.1. We start the system in the random fixed point ao. In this case , we do not need time for attractivity, as due to the stationarity of ao and Theorem 3.2 u(O) := ao already fulfils the assumptions of Theorem 3.4 . REMARK 4.1. We do not use uniformity in the initial condition. Hence, we can only prove results for random fixed points, and not for random set attractors but it would be an interesting result , whether we have
on time intervals of order O(c- 2 ) with high probability. REMARK 4 .2 . The restriction to random fixed points still covers several cases. For example for dissipative nonlinearities c 2 A + F (e.g. -c 2 u - u 3 ) it is well known, that the random attractor is just a single random fixed point. For non-dissipative nonlinearities (e.g. c 2 u - u 3 ) it is in most cases completely open what the topology of the random aitracior is. But, nevertheless, in many examples of non-trivial random attractors for SPDEs these attractors contain random fixed points. If the attractor for the amplitude equation is a single stable fixed point a*, which is exponentially attracting, then we can proof a much stronger result. We suppose the following . ASSUMPTION 5 . Suppose that the random dynamical system generated by the amplitude equation (2.4) has a unique random fixed point a* that is exponentially attracting in p-th mean. This means, for any p > 0 there are constants 6 > 0 and M';» 0 such that
for any solution a of (2.4) .
52
DIRK BLOMKER AND MARTIN HAIRER
For simplicity, we will rescale the equation to the slow time-scale T = €2t. Consider the rescaling v(T) = €-l u(Tc 2), which is a solution of (4.2)
aTV =
€-2 Lv
+ Av + F(v) + aTQW ,
where W is just a rescaling of W. Let a be a solution of the amplitude equation with j3 = PcQW and let 'l/Je be the rescaled Ornstein-Uhlenbeck process
Consider now the random dynamical system generated by the triple (v, a, €'l/Je). It is obvious that random fixed points of this equation are just rescaled versions of random fixed points for the original system of equation. We can prove the following theorem. THEOREM 4 .2. Suppose Assumptions 1, 3, 4, and 5 with random fixed point a* are true. Let v* be any fixed point of the rescaled equation (4.2) and denote by 'I/J; the rescaled stationary 0 U-process. Then for any small K, E (0,1) and any p > 0 there is a constant C such that lE ( Ilv* - a* - €'I/J; II~ )
I/ P
::; C€2-t<.
Proof For the proof consider first the projection to N . Let a be a solution of (2.4) with €-1 Pcv*. Using stationarity, 1E(llPcv* - a*II~)l/p = 1E(llPcv*(t9T') - a*(t9T')II~)l/p ::; 1E(llPcv*(t9T') - a(T)II~)l/p + 1E(lIa(T) - a*(t9T ')II~ )l / p ::; C€2-t<
+ M ae- oTIE(IIPcv* _
a*II~ )l / p ,
where we used the approximation result and Assumption 5. Given p and oT < ~ . This yields the K" we can take T sufficiently large such that Maefirst part of the proof. The second part is completely analogous using the exponential stability of the Ornstein-Uhlenbeck process . 0 REMARK 4 .3. We can also establish an analog of the approximation result, which implies
This is not an immediate consequence of a rescaled version of Theorem 3.4, we have to modify this slightly to allow for more general initial conditions for the amplitude equation.
AMPLITUDE EQUATIONS FOR SPDE
53
5. Approximation of the invariant measure. In this section, we review the results obtained in [7] on the invariant measure of (2.1). Recall that if !Pt denotes the stochastic flow generated by the solutions to (2.1), then a measure J1.* on 'H is called an invariant measure if (5 .1)
for every t > 0 and for every Borel set A c 'H. For a general survey on invariant measures for SPDEs see [19]. The results of this section will rely on a very mild non-degeneracy condition for the stochastic forcing in (2.1) (cf. Assumption 4). This is the content of the next assumption. Note that this condition could be relaxed for some examples. ASSUMPTIO N 6. The operator PcQQ*Pc is invertible as -on operator from N to N . REMARK 5 .1 . This assumption ensures that the amplitude equation (2.4) has a unique invariant measure, and that this invariant measure has a Coo density with respect to the Lebesgue m easure. Furthermore, we have exponential convergence of distribut ions of solutions to the invariant measure (see e.g. [33]). There are many situations where this assumption also ensures the existence of a unique in variant measure for the original SPDE, see for example the recent works [30, 32, 22, 24]. We will make use of two different norms to measure the distances between invariant measures . The first is the Wasserstein norm (also called Kantorovich distance), which is defined as the dual norm to the Lipschitz norm (5.2)
11 4>IIL = sup { 14>(x)1 , 14>(x) - 4>(y)I} Ilx-yll ' x ,y E'H
Le. one has
(5.3) This will give us convergence for prob abilities and , as we have uniform bounds on arbitrary moments (cf. Theorem 3.2), convergence of moments and other statistical quantities. The second norm is the total variation norm 1 ·IITv, which is defined as the dual norm to th e Loo-norm. Since 11 4>IIL 2 11 4>1100 , the total variation norm is stronger th an th e Wasserstein norm. Note that the Wasserstein norm depends strongly on the metric that equips the underlying spa ce, whereas the total variation norm is indep endent of that metric. For exa mple, the Wasserstein norm between two Dirac measures 8x and 8y is given by min{l , Ilx - YII}, whereas 118x - 8yll T V is given by 1 if x -=1= y and 0 otherwise. (Actually, one can show that the total variation norm is equal to
54
DIRK BLOMKER AND MARTIN HAIRER
the supremum over all possible metrics of the corresponding Wasserstein norms. See [38] for a beautiful discussion of the relationship and properties between various metrics on probability measures .) Before we state our results, we introduce one more notation. Similar to the proof of Theorem 4.2, we will rescale the solutions of (2.1) by e such that they are concentrated on a set of order 1 instead of a set of order c. Furthermore, we will rescale the equation to the slow time-scale T = tc 2 . We denote by J.L; the invariant measure of the rescaled version of (2.1). We furthermore denote by the invariant measure for the pair of processes (a, c'ljJ). Note that v; depends on e by the rescaling of 'ljJ and by the fact that equations (2.4) and (2.5) are coupled through the noise, but do not live on the same time scale. However, the marginal of on N is independent of e and its marginal on S depends on e only through the trivial scaling of c'ljJ. We denote these two marginals by and With these notations, our result in the Wasserstein distance is the following: THEOREM 5.1. Let Assumptions 1, 3, 4, and 6 hold. Then, for every K > 0, one has
v;
v;
vz
v:.
(5.4) 5.2. Actually, one also has IIJ.L; - v;IIL = O(c 2- K ) , but the above formulation is mo re interesting, since vZ and v: can be characterised explicitly, whereas v; can not, unless the covarian ce operator Q is blockdiagonal with respect to the splitting 1-£ = N EB S . Idea of proof Denote by Qt the Markov transition semigroup (acting on measures) associated to the rescaled version of (2.1), and by Pt the transition semigroup associated to the evolution of (a(t) ,c'ljJ(c-2t)) . Then, the main ingredient for the proof of Theorem 5.1 is that there exists a time T such that, for every pair (J.L, v) of probability measures with finite first moment, one has REMARK
In order to prove (5.5), one uses the strong contr action properties of the linear dynamic in S and that the strong mixing properties of the nondegenerate noise in N . Once (5.5) is established, the proof of Theorem 5.1 follows in a rather straightforward way. One first obtains from Theorem 3.4 that IIJ.L~
- v;IIL :::;
+ IIPTJ.L~ - PTv;liL - 1/; IlL + O(c 2 ) ,
IIQTJ.L~ - PTJ.L~IIL
:::; O(c 2 -
K )
1 + 211J.L~
and therefore IIJ.L; -v;IIL = O(c 2 - K ) . The bound 111/; -vZ0v:liL = O(c 2 - K ) is then obtained by using the smoothness of the density of vZ with respect to
AMPLITUDE EQUATIONS FOR SPDE
55
the Lebesgue measure, combined with the separation of time scales between 0 the dynamics on N and on S. The first result in the total variation norm only considers the marginals of the invariant measures on N. THEOREM 5.2 . Let Assumptions 1, 3, 4, and 6 hold. Then, for every r: > 0, one has
(5.6) Idea of proof We combine the smoothing properties of PcPtPc with the result previously obtained in Theorem 5.1 to show that
(5.7)
v:
C
2-1<
I/PT/L; - PT IlL S ~
for all T E (0,1].
Then, we use Girsanov 's theorem to show that
Combining both estimates and optimising for T yields the result. 0 REMARK 5 .3. Th e bound (5 .6) is not always optimal. For exam ple, when L and A are selfadjoint, Q is the iden tity and F is the gradient of .a potential V, the rescaled invarian t m easure /L~ for (2.1) can formally be written as
(5.9)
/L;(du) = exp ( ~( u, Au) - V(u))/L6(du) ,
where /Lb is the product of the Gaussian measure with covariance E 2 L;' on S and the Lebesgue measure on N . This explicit expression allows on e to show that the dens ity of P; /Le. has derivatives of all orders and that these derivatives are all of order 1. Th is knowledge can be com bin ed with Theorem 5.1 to show that in this case IlPc* /L; -vZIITV = O( E2- 1< ). However, this argum ent fails completely if, for example, PsQPs = O.
Our last result on the convergence of the invariant measures of the amplitude equation measures the distance between /L; and vZ® v: in the total variation norm. This however requires to impose a much stronger non-degener acy assumption on the noise. ASSUMPTION 7. Let 0: be as in Assumption 2. There exists a constant > 0 such that, for all , E [0, 'a], :F : (11.,)3 ~ 11.,-0: and A : 11.' ~ 'H'-O: are continuous. Furth ermore, the operator Q-1 is con tinuous from 11.'0-0: to 11. and for som e a E [O,~) we have 11(1- L)'o- ciQIIHs(x) < 00 . THEOREM 5 .3 . Let Assumptions 1, 3, 4, and 7 hold. Th en, for every r: > 0, one has
'A
(5.10)
56
DIRK BLOMKER AND MARTIN HAIRER
Idea of proof We denote by to the linear system (5.11)
i\
the transition probabilities associated
du = c- 2Ludt + QdW(t) .
It is then possible to show as above by Girsanov theorem that (5.12) Furthermore, the fast relaxation of the S-component of the solutions to (5.11) toward its equilibrium measure, combined with the fact that the N marginals of J..t; and of IJ; are close by Theorem 5.2, allows to show that II'PTJ..t; - IJ~ Q9 1J: IITV :S Cc, provided that T» c2 • The result then follows 0 by choosing T of the order c 2 - o for some small value of 8. 6. What is so special about cubic nonlinearities? Cubic nonlinearities are not special, we can extend the method to a lot of different types of nonlinearities. Suppose we have a multi-linear nonlinearity, which is homogeneous of degree n. Then the noise strength in the SPDE (2.1) should be changed to c(n+l) /(n-l) instead of c 2. Now with the ansatz
and a similar formal calculation as in section 2, we derive the amplitude equation
which now contains also a nonlinearity that is homogeneous of degree n. We can verify this result rigorously. After minor changes the local theorems immediately carry over to these kinds of equations. For example for stable odd nonlinearities at least the order 1 approximation (local and global) is completely analogous . The local approximation results also carry over to even nonlinearities, but one problem for global results is that we do not have nonlinear stability of the equations. In some cases, we can however get global results for even nonlinearities, if we already have good a-priori bounds for the solutions. But the main problem with quadratic nonlinearities B (u) = B (u, u) is that in many examples PcB(a) == for a E N. In this case, the previously mentioned result will give us only the linearisation, meaning that we still look at solutions that are too small to capture the nonlinear features of the equation. To illustrate this problem , we will briefly discuss Burgers equation, which is given by OtU = o;u + J..tU - uOxu + (j€~ . For periodic boundary conditions and J..t = O(c 2 ) we get N = span{l} but now already B(l) = 0. If we consider Dirichlet boundary conditions on [0,11"], for example, then the linear instability arises for J..t = 1 + O(c 2 ) .
°
57
AMPLITUDE EQUATIONS FOR SPDE
Furthermore, N = span{sin} and Be(sin) = 0, where we used the shorthand notation B; = PeB and B, = PsB. There are numerous examples in the physics literature of equations with quadratic nonlinearities and the same property, as described above. One example is the growth of rough amorphous surfaces. See for example [6] and the references therein. Another example is the celebrated KuramotoSivashinsky equation, but the probably most important example is the Rayleigh-Benard problem (see e.g. [23] or [15]) which is the paradigm of pattern formation in convection problems . If we want to take into account nonlinear effects, we then have to look at the coupling of the slow dominant modes to the fast modes. This was done in [8] for the local result. Let us now briefly comment on these results. Consider an equation of the type
with Be(a,a) = 0 for a EN, where B is symmetric and bilinear. We make the ansatz
with a E N (£) and 'l/J E PsX. This yields in lowest order in e the following system of formal approximations. First of order 0(£2) on the fast timescale t in PsX.
Secondly of order (6.2)
£3
Bra(T)
=
in N (£) on the slow time-scale T = £2t
Aea(T) + 2Be(a(T ),'l/J(e- 2T» + Pe€(T),
where f.(T) = £-1~(£-2T) is a rescaled version of the noise. These equations are on one hand a dominating equation (6.2) on a slow time-scale coupled to an equation (6.1) on the fast time-scale. Equations with a similar structure are treated in [3] for stochastic ODEs, or in [16, 17] where tracers in a fast moving velocity field are considered . The aim is now to get an effective equation for the slow component completely independent of the fast modes. First rescale (6.1) to the slow time-scale T by 'l/J(t) = (£2t). Hence,
As L; is invertible on PsX, we get in lowest order of e that (T) = -£;lB s(a(T),a(T)) . This together with (6.2) establishes a single approximation equation.
(6.3)
fJra(T) = Aea(T) - 2Be (a(T), £-;1e, (a(T), a(T)) )
+ pi(T) ,
58
DIRK BLOMKER AND MARTIN HAIRER
Surprisingly, this equation involves a cubic nonlinearity, although the nonlinearity in the original equation was quadratic. The main results of [8] show that these formal calculations can be made rigorous in the sense of Theorems 3.1 and 3.3.
REFERENCES [1] L. ARNOLD, Random dynamical systems, Springer Monographs in Mathematics. Springer, Berlin, 1998. [2] B . AULBACH, Continuous and discrete dynamics near manifolds of equilibria, Lecture Notes in Math., 1058, Springer, Berlin, 1984. [3] N. BERGLUND AND B . GENTZ, Geometric singular perturbation theory for stochastic differential equations, J . Differential Equations 191(1): 1-54 (2003) . [4] D . BLOMKER, Amplitude equations for locally cubic non-autonomous nonlinearities, SIAM J. Appl. Dyn. Sys., 2(2) : 464-486 (2003). [5J D . BLOMKER, S . MAIER-PAAPE, AND G . SCHNEIDER, The stochastic Landau equation as an amplitude equation , Discrete and Continuous Dynamical Systems, Series B , 1(4) : 527-541 (2001). [6] D. BLOMKER, C .GUGG, AND M . RAIBLE, Thin-mm-growth models: roughness and correlation functions, European J . Appl. Math., 13(4): 385-402 (2002). [71 D . BLOMKER AND M. HAIRER, Multiscale expansion of invariant measures for SPDEs, to appear in Commun. Math. Phys., 2004. [8] D . BLOMKER, Approximation of the stochastic Rayleigh-Benard problem near the onset of instability and related problems, Submitted for publication, 2003. [9] D . BLOMKER, M . HAIRER, AND G . PAVLIOTIS, Stochastic amplitude equations in large domains, In Preparation, 2004. [lOJ Z. BRZEZNIAK AND S . PESZAT, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process , Studia Math. 137(3): 261-299 (1999) . [l1J Z. BRZEZNIAK AND S . P ESZAT, Strong local and global solutions for stochastic Navier-Stokes equations . Infinite dimensional stochastic analysis (Amsterdam, 1999), pp. 85-98, Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. , 52, R . Neth. Acad. Arts ScL , Amsterdam (2000). [12] P . COLLET AND J .P . E CKMANN, Instabilities and fronts in ext end ed systems, Princeton Univ. Press, Princeton, NJ, 1990. [13] P . COLLET AND J .P. ECKMANN, Th e time dependent amplitude equation for the Swiit-Hohenoetg problem , Comm. Math. Phys., 132(1): 139-153 (1990) . [14] H . CRAUEL, A . DEBUSSCHE, AND F . FLANDOLI, Random attractors, J . Dynam. Differential Equations, 9(2): 307-341 (1997). [15] M .C . CROSS AND P .C . HOHENBERG, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 : 851-1112 (1993). [16J G . PAVLIOTIS AND A . STUART, White noise limits for inertial particles in a random field. Preprint (2003). [17] G . PAVLIOTIS AND A . STUART, Ito versus Stratonovich white noise limits. Preprint
(2003). [18] G . DA PRATO AND J . ZABCZYK, Stochastic Equat ions in Infinite Dimen sions, Cambridge University Press, 1992. [19] G . DA PRATO AND J . ZABCZYK, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, 229, Cambridge University Press (1996). [20J J . Du AN AND V .J. ERVIN, On nonlinear amplitude evolution under sto chastic forcing, Appl. Math. Comput. 109(1): 59-65 (2000). [21J J. DUAN, K. Lu , AND B . SCHMALFUSS , Invariant manifolds for stochastic partial differential equations , The Annals of Probability, to appear.
AMPLITUDE EQUATIONS FOR SPDE
59
[22] W. E AND D. LIU, Gibbsian dynamics and invariant measures for stochastic dissipative PDEs, Journal of Statistical Physics, 108: 4773-4785 (2002) . [23] A.V . GETLlNG, Reyleigh-Betierd Convection - Structures and Dynamics, World Scientific Press, 1998. [24] M. HAIRER AND J .C . MATTINGLY, Ergod icity of the 2D Navier- Stokes Equations with Degenerate Sto chastic Forcing. Preprint, 2004. [25] H. HAKEN, Synergetics. An introduction. Nonequilibrium phase transitions and self- organization in physics, chemistry, and biology, Springer Series in Syn ergetics, Vol. 1. Berlin et c.: Springer, 1983. [26] D. HENRY, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Berlin etc .: Springer, 1981. [27] P .C . HOHENllERG AND J .B . SWIFT, Effect s of additive noise at the onset of Rayleigh-Benard convection, Physical Review A, 46 : 4773-4785 (1992). [28] B.B . KING , O. STEIN , AND M. WINKLER, A fourth-order parabolic equation modeling epitaxial thin JJlm growth, J . Math. Anal. Appl. , 286(2) : 459-490 (2003) . [29] R. KUSKE, Multi-scale analysi s of noise-sensitivity near a bifurcation, Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26-30, August 2002 (eds . N. Sri Namachchivaya and Y.K. Lin) , Kluwer , 2002. [30J S.B . KUKSIN AND A. SHlRIKYAN , A Coupling Approach to Randomly Forced Nonlinear PDE's. I, Commun. Math . Phy s., 221 : 351- 366 (2001) . [311 A. LUNARDI, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and th eir Applications, 16 , Birkhauser Verlag, Basel , 1995. [32] J .C . MATTINGLY, Exponential convergence for the stochastically forced NavierStokes equations and other partially dissipative dyn amics, Commun. Math . Phys. , 230 : 421-462 (2002). [33] S.P . MEYN AND R.L. TWEEDlE, Markov Chains and Stochastic St ability, Springer , New York , 1994. [34] A. MIELKE, G. SCHNEIDER, AND A. ZIEGRA , Comparison of inertial man ifolds and application to modulated systems, Math. Nachr., 214 : 53-69 (2000). [35] A. MIELKE AND G . SCHNEIDER, Attractors for modulation equations on unbounded domains - exist ence and compari son, Nonlinearity, 8 : 743-768 (1995). [361 S. MOHAMMED , T. ZHANG, AND H. ZHAO, The Stable Manifold Theorem for Semilinear Sto chastic Evolution Equations and Stochastic Partial Differential Equations . Part I: The Stoch astic Semiflow , preprint (2003). [37] A . PAZY, Sem igroups of Lin ear Operators and Application to Partial Differential Equations, Springer, 1983. [38] S. RACHEV, Probability metrics and the stability of stochastic models, John WHey & Sons Ltd., Chichester, 1991. [39] M. SCHEUTZOW, Comparison of various concepts of a random at tractor: a case study, Arch . Math. (Basel) ,18(3) : 233-240 (2002). [40] G . SCHNEIDER, Bifurcation theory for dissipative systems on unbounded cylindrical domains-an introduction to the mathematical th eory of modulation equations, ZAMM (Z. Angew. Math. Mech.) 81(8) : 507-522 (2001). [41J B. SCHMALFUSS, Measure attractors and random attractors for stochastic partial differential equations, Stochastic Anal. Appl. , 11(6): 1075-1101 (1999).
ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS VENA PEARL BONCOLAN-WALSH', JINQIAO DUAN*, HONCJUN
cxot,
TAMAY OZCOKMENt , PAUL FISCHER§, AND TRAIAN ILIESCU'
Abstract. We study a coupled deterministic system of vorticity evolution and salinity transport equations, with spatially correlated white noise on the boundary. This system may be considered as a model for gravity currents in oceanic fluids . The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain an asymptotic estimate of enstrophy evolution, and then show that the system is ergodic under suitable conditions on mean salinity input flux on the boundary, Prandtl number and covariance of the noise . Key words. Random dynamical system, stochastic geophysical flows, enstrophy, climate dynamics, ergodicity. AMS(MOS) subject classifications. Primary 60H15 ; Secondary 86A05 , 34D35 .
1. Geophysical background. A gravity current is the flow of one fluid within another driven by the gravitational force acting on the density difference between th e fluids. Gravity currents occur in a wide variety of geophysical fluids. Oceanic gravity currents are of particular importance, as they are intimately related to the ocean's role in climate dynamics. The thermohaline circulation in the ocean is strongly influenced by dense-water formation that takes place mainly in polar seas by cooling and in marginal seas by evaporation. Such dense water masses are released into the large-scale ocean circulation in the form of overflows, which are bottom gravity currents .
We consider a two-dimensional model for oceanic gravity currents, in terms of the Navier-Stokes equation in vorticity form and the transport equation for salinity. The Neumann boundary conditions for this model involve a spatially correlated white noise due to uncertain salinity flux at the inlet boundary of the gravity currents. In the next section, we present the model and reformulate it as a random dynamical system, and then discuss the eocycle property and dissipativity of this model in §3 and §4, respectively. Main results on random 'Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA (bongvenat i t edu: duanlUiit.edu) . tDepartment of Mathematics, Nanjing Normal University, Nanjing 210097, China (gaohjenjnu . edu. en) . tRSMAS /MPO, University of Miami, Miami, Florida, USA (tamaylUrsmas .miami. edu) . §Argonne National Laboratory, Argonne, Illinois, USA (fiseherlUmes.anl.gov) . 'Mathematics Department, Virginia Tech, Blacksburg, VA 24061, USA (ilieseulUealvin .math . vt . edu) . i
61
62
VENA PEARL BONGOLAN-WALSH ET AL.
attractors, enstrophy and ergodicity are in §5. Enstrophy is one half of the mean-square spatial integral of vorticity. Ergodicity implies that the time average for observables of the dynamical system approximates the statistical ensemble average , as long as the time interval is sufficiently long.
2. Mathematical model. Oceanic gravity currents are usually down a slope of small angle (order of a few degrees) . We model the gravity currents in the downstream-vertical plane , and ignore the variability in the cross-stream direction. This is an appropriate approximate model for, e.g., the Red Sea overflow that flows along a long narrow channel that naturally restricts motion in the lateral planar plane [11] . In fact, we will ignore small slope angle and the rotation, both affect the following estimates nonessentially, i.e., our results below still hold with non-essential modification of constants in the estimates and in the conditions for the ergodicity. Thus we consider the gravity currents in the downstream-vertical (x, z )-plane. It is composed of the Boussinesq equations for ocean fluid dynamics in terms of vorticity q(x, z, t), and the transport equation for oceanic salinity S(x,z,t) on the domain D = {(x ,z): 0::; x,z::; I} :
qt + J(q ,'IjJ) =tJ.q - RaoxS, (1)
1
s, + J(S, 'IjJ) =Pr tJ.S,
where
q(x, z, t) = -tJ.'IjJ is the vorticity in terms of stream function 'IjJ , Pr is the Prandtl number and Ra is the Rayleigh number. Moreover, J(g, h) = gxhz-gzhx is the Jacobian operator and ~ = oxx + ozz is the Laplacian operator. All these equations are in non-dimensionalized forms. For the simplicity, we let Pr = 1. Note that the Laplacian operator tJ. in the temperature and salinity transport equations is presumably oxx+~152ozz with 8 being the aspect ratio, and KH, KV the horizontal and vertic~ diffusivities of salt, respectively. However, our energy-type estimates and the results below will not be essentially affected by taking a homogenized Laplacian operator tJ. = oxx + ozz. All our results would be true for this modified Laplacian. The effect of the rotation is parameterized in the magnitude of the viscosity and diffusivity terms as discussed in [19]. The fluid boundary condition is no normal flow and free-slip on the whole boundary
'IjJ = 0, q = O. The flux boundary conditions are assumed for the ocean salinity S . At the inlet boundary {x = 0, 0 < z < I} the flux is specified as:
(2)
oxS = F(z)
+ w(z, t),
ENSTROPHY AND ERGODI CITY OF GRAVIT Y CURRENTS
63
with F (z) being the mean freshwater flux , and the fluctuating par t w(z, t ) is usu ally of a shorte r t ime scale t han t he response t ime scale of t he oceanic mean salinity. So we neglect t he a utocorr ela t ion t ime of this fluctuating process and t hus ass ume that the noise is white in t ime. The spatially correlat ed white-in-time noise w(z , t) is desc ribed as the generalized t ime derivative of a Wi en er pro cess w(z, t) defined in a pr ob ability space (D, IF, JPl) , with mean vect or zer o and covar ia nce operator Q. On t he outlet boundary {x = 1, 0< z < I}:
At the t op boundar y z
= 1, and at t he
bottom bo undary z
= 0:
This is a system of deterministic partial differ ential equat ions with a stochastic boundary condit ion.
3. Cocycle property. In this section we will show that (1) has a unique solution, and by reformulating t he model, we see it defines a eo cycle or a random dynamical system. For t he followin g we need some tools from t he t heory of par ti al different ial equations . Let Wi (D) be t he Sobolev space of fun ct ions on D wit h first generalized der ivative in L 2 (D ), t he function space of square int egr able functions on D with norm and inn er produ ct 1
IIullL2 = ( l
lu (x )12dD)
2,
(u, V)L 2
=
1
u(x)v(x)dD,
u, v E L 2(D ).
The space Wi (D ) is equipped wit h t he norm
1;Jotivated by the zero-boundary condit ions of q we also introduce the space W~(D) which contains ro ughly spe aking func tions wh ich are zero on the boundary oD of D. This space ca n be equipped with the norm
(3) Simil arl y, we can define fun cti on spaces on t he int er val (0, 1) denoted by L 2 (0, 1). Another Sob olev space is given by Wi( D) which is a subs pace of Wi( D) consisting of fun cti ons h such t hat hdD = 0. A norm equivalent to t he Wi-norm on Wi (D ) is given by t he right han d side of (3) . For t he subs pace of functions in L 2(D ) having t his prop erty we will de note as £ 2(D ).
ID
64
V ENA PEARL BONGOLAN-WALSH ET AL.
We reformulate th e above stoc hastic initi al-boundary value problem into a random dynamical syste m [1] . For convenience, we introduce vector not ati on for unknown geophysical quantities u = (q, S).
(4)
Let w be a white noise in L 2 (0, 1) with finite trace of th e covariance operator Q, and t he Wiener process w(t) be defined on a probability space
(n, IF, JP). Now we choose an appro priate phase spa ce H for t his syste m. We assume th at t he mean salinity flux F E L 2(0 , 1). Not e that
:t
l
1 1
SdD
=
[F (z) + w(z, t) ]dz = constant .
It is reasonable (see [5]) to assume that
1 1
(5)
[F( z) + w(z, t)]dz = 0,
and thus JD SdD is constant in time and , without loss of generality, we may assume it is zero (otherwise, we subt ract thi s constant from S):
l.
SdD=O.
Thus S E L2 (D ), and we have t he usual Poincare inequality for S. Define the phas e space
We rewrite th e coupled syste m (1) as:
(6)
du dt + Au = F 1 (u) + F2 (u),
u(O ) = Uo E H ,
where
Au =
F1 (u)[x , z] =
-~q (
)
1
- - ~S
,
Pr
- J( q, 'I/J) ) [x , z], ( - J( S, 'I/J)
and
F2 (u)[x, z] = (
°
Ra(-oxS ) )
[x, z].
ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS
65
The boundary and initial conditions are: q = O,onoD ,
oS on (7)
=
0, on oD\{x = 0, 0< z < I} ,
oS
ox = F(z) +w(z,t), on {x = 0, 0 < z < I},
u(O) = uo = (
t ),
where n is the out unit normal vector of Bl), The system (6) consists of deterministic partial differential equations with sto chastic Neumann boundary conditions. We now transform the above system (6) into a system of random partial differential equations (Le., evolution equations with random coefficients) with homogeneous boundary conditions, whose solution map can be easily seen as a cocycle. Thus we can investigate this dynamics in the framework of random dynamical systems [1] . Note that we have a nonhomogenous stochastic boundary condition for salinity S , so the first step is to homogenize this boundary condition. To this end, we need an Ornstein-Uhlenbeck stochastic process solving the linear differential equation
(8) with the following same boundary conditions as for S, and zero initial condition
OxTJ1(t,0,Z ,W) = F(z)
+ w(z , t) ,
OxTJ1(t ,I,z,w) = 0, OzTJ1(t, x ,0,w) = 0, (9)
OzTJ1(t ,x,I ,w) =0, TJl(O ,X,Z,w) = 0,
1
TJ1dD = 0.
LEMMA 3.1. Suppose that the covariance Q has finite trace : tr L2 Q < Then the Ornstein-Uhlenbeck problem (8)-(9) has a unique stationary solution in L 2(D) generated by
00 .
66
VENA PEARL BONGOLAN-WALSH ET AL.
In fact , we can write down the solution Zabczyck [13, 12] as
(10)
T}l(t , x , z,w) = (-Ll)
l
T}l
following Da Prato and
t
S(t - s)N(X)ds
where I is the identity operator in L2(D) , and N is the solution operator to the elliptic boundary value problem Llh - )"h = with the boundary conditions for h the same as T}l, that is oh/an = X on aD with hdD = 0, where n is the unit outer normal vector to aD and X is
°
ID
x= ( F(ZlTZ,Sl ) .
Here X is chosen so that this elliptic boundary value problem has a unique solution. Since hdD = 0, we can choose X = 0. Moreover, S(t) is a strongly continuous semigroup , symbolically, eAt, that is, the generator of S(t) is Ll. Now we are ready to transform (6) into a random dynamical system in Hilbert space H. Define
ID
(11)
~l
T}(t ,x,z,w) = (
)
and recall
Let (12)
v:= U
-
T} .
Then we obtain a random partial differential equation
v(o) = Vo E H ,
(13)
where vx(t,O,z,w), vx(t ,I , z,w), vz(t , x ,O,w) and v z(t ,x,l,w) are now all zero vectors.i.e., homogenous boundary conditions, and the initial condition is still the same as for u
v(O ,x,z,w) = (
t: ).
However, because of the Jacobian, we will have to do nonlinear analysis on (13) to resolve v.
ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS
67
We introduce another space
For sufficiently smooth functions v = (ij, S) , we can calculate via integration by parts
(Au, V ) H =
(14)
1
\lq. \lijdD 1
r
-
+ Pr ) D \lS . \lS dD since now, we have only homogenous boundary conditions. Hence on the space V we can introduce a bilinear form a(.,.) which is continuous, symmetric and positive
a(u, v) =
1
\lq. \lijdD +
1vs vs.u:
This bilinear form defines a unique linear continuous operator A : V -; V' such that (Au, v) = a(u, v). Recall
Ft(u)[x, z] = (
=;~~,~~
)
[x,z]
and
) F2 (u)[x, z] = ( Ra(-oxS) 0 [x, z]. LEMMA 3 .2. The operator
Fi : V -; H is continuous. In particular,
we have
(Fl(U),U ) = O. Proof We have a constant
Cl
> 0 such that
(15) for any q E Wi (D) which follows st raight forwardly by regularity properties of a linear elliptic boundary problem. Note th at wt is a Sobolev space with respect to the third derivatives. Hence we get :
IIJ(S,1/')11£2::; sup (lox1/'(x,z)1 + loz1/'(x,z)l) x (x ,z )ED
X
(l'OxS(X, z)1
+ lozS(x,z)ldD) .
68
VENA PEARL BONGOLAN-WALSH ET AL.
The second factor on the right hand side is bounded by
On account of the Sobolev embedding Lemma, we have some positive constants C2, C3 such that sup (Iax 'l/J(x, z)/ + laz'l/J(x, z)l) :Sc211'V'l/J 1 1wi (D ) :Sc31Iqllwi(D) :Sc31Iullv. (x ,z)ED Hence we have a positive constant
C4
such that
for u E V . We now show that
(J(S, 'l/J ),S) =
o.
We obtain via integration by parts
LaxSaz'l/J SdD - LazSax'l/J SdD = - La;zS'l/J SdD + La;xS'l/JS dD -LaxS'l/JazS dD
+
razS'l/JaxSdD + r
}D
J(O ,I)
axS'l/JSI~~5dx -
r
J(O,I)
azS'l/JSI~~5dz = 0
because 'l/J is zero on the boundary Bl), This relation iso true for a set of sufficiently smooth functions 'l/J , S which are dense in W~(D) x Wi(D) . By the continuity of F I , as just shown in Lemma 3.2, we can extend this o . property to W~(D) x Wi(D). 0 LEMMA 3.3 . The following estimate holds
for some positive constant cs . Proof By simple calculation, the proof is obtained.
0 We have obtained a differential equation without white noise but with random coefficients. Such a differential equation can be treated samplewise for any sample w . We are looking for solutions in v E C([O, T]; H) n L 2(0, T; V) ,
for all T > O. If we can solve this equation then u := v + 7J defines a solution version of (6). For the well posedness of the problem we now have the following result.
ENSTROPHY AND ERGODICITY OF GRAVI TY CURRENT S
69
3.4 (Well-Posedness) . For any time r > 0, there exists a uni que solution of (13) in C( [O, r ]; H ) n £ 2(0, r ; V). In particular, the solution mapping THEOREM
IR+ x
nx
H 3 (t ,w , vo) ---+ v(t) E H
is measurable in its arguments and the solution mapping H 3 Vo ---+ v(t ) E H is continu ous. Proof By the properties of A and Ft (see Lemm a 3.2), t he random differenti al equation (13) is essentially similar to t he 2 dim ensional Navier Stokes equation. Note th at F2 is only an affine mapping. Hence we have 0 existence and uniqueness and the above regulari ty assertions. On account of t he transform ation (12), we find t hat (6) also ha s a uniqu e solution. Since the solution mapping IR+ x n x H 3 (t ,w, vo)
---+
v(t ,w , vo) = : 4?(t,w,vo ) E H
is well defined, we can introdu ce a random dyn amical system . On n we can define a shift operator ()t on th e paths of the Wiener process t hat pushes our noise:
w( ·,{hw) = w( · + t ,w) - w(t,w)
for t E IR
which is called the Wiener shift. Then {()t} tE IR forms a flow which is ergodic for the probability meas ure JP>. The properties of t he solution mapp ing cause the following rela tions
4?(t + r ,w ,u) = 4?(t , ().,w, 4?(r ,w ,u))
for
t, r 2: 0
4?(O,w, u) = u for any wE n and u E H. This prop erty is called t he eocycle proper ty of 4? which is import ant to study the dynamics of random syste ms. It is a generalization of t he semigroup property. The eocycle 4? toget her with the flow () forms a random dynamical system. 4. Dissipativity. In t his section we show t hat the random dynamical syste m (13) for gravity curre nts is dissip ative, in the sense t hat it has an absorbing (random) set . This means that the solut ion v is cont ained in a particular region of t he phas e space H after a sufficiently long tim e. This dissip ativity will help us to obtain asymptoti c est ima tes of the enst rophy and salinity evolution. Dynamical properties that follow from this dissipat ivity will be considered in the next sect ion. In par ticular , we will show that the system has a ra ndom at t rac tor , and is ergodic und er suit able conditions. We introduce the spaces
fI = £2(D ) V = Wi(D).
70
VENA PEARL BONGOLAN-WALSH ET AL.
We also choose a subset of dynamical variables of our system (1)
v=
(16)
S-
TJl .
To calculate the energy inequality for V, we apply the chain rule to We obtain by Lemma 3.2
Ilvllk.
~ Il vlll + 211V' v11L
(17)
=2(J(TJl, 1jJ), v). The expression V'v is defined by (V' x,zv). We now can estimate the term on the right hand side. By the Cauchy inequality, integration by parts and Poincare ine9uality AlllqllL~ "V'qIIL~ for q E W~(D) and A211iillL2 < IIV'iiIIL2 for v E Wi(D), we have
s
(18) For q, we have the following estimate
(19) From (18) and (19), we have
(20)
:t (2 11v111 + Ra~A~ Ilqll2) + IIV'vIIL +
(Ra;A~ -
2 2,xi11TJl1 }1V'qI12
$
~~ IITJlI1 2.
DEFINITION 4 .1. A random set B = {B(W)}wEO consisting of closed bounded sets B(w) is called absorbing for a random dynamical system ep if we have for any random set D = {D(W)}WEO, D(w) E H bounded, such that t ---+ SUPyED(O,w) IlyllH has a subexponential growth for t ---+ ±oo
(21)
ep(t,w,D(w)) C B(Btw) ep(t, B_tw, D(B_tw)) C B(w)
for for
t2to(D,w) t 2 to(D, w).
B is called forward invariant if ep(t,w,uo)EB(Btw)
ifuoEB(w)
fort20.
Although v is not a random dynamical system in the strong sense we can also show dissipativity in the sense of the above definition. LEMMA 4.2. Let ep(t , w, vo) E H for Vo E H be defined in (6), and 1
Ra2A~ -
2
2
2A 1lEIITJlI > O.
ENSTROPHY AND ERGODICITY OF GRAVITY CU RRENTS
71
Then the closed ball B(O,RI(w)) with radius RI(w) =
2[°00 e"'T:~ 1I1']1112dT
is forward invariant and absorbing. The proof of this lemma follows by int egration of (20) . For the appli cations in th e next section we need that the elements which are contained in th e absorbing set satisfy a particular regularity. To this end we introduce th e functi on space
:= {u EH : Ilull; := IIA~ ull~ < oo}
'W
where s E R The operator As is th e s-th power of th e positive and symmetric operator A. Note that these spaces are embedded in the Slobodeckij spaces HS, s > 0. The norm of thes e spaces is denoted by I1 . 1I n-. This norm can be found in Egorov and Shubin [7]' P age 118. But we do not need this norm explicitly. We only mention that on re th e norm I . lisof HS is equivalent to the norm of re for < s, see [8] . Our goal is it to show that v(l, w , D) is a bounded set in 'H" for some s > 0. This property causes th e complete continuity of th e mapping v(l, w, .). We now derive a differential inequ alit y for tllv(t )II;. By the chain rule we have d d
°
dt (t [[ v(t)II; ) =
Il v(t)ll; + t dt Il v(t)II ;·
Note that for the emb edding const ant
Cs
between H' and V
1Ilvll;ds::; 1Il vll~ds t
t
c;
for s ::; 1
such th at the left hand side is bound ed if the initial condit ions Vo are contained in a bounded set in H . The second term in the above formula can be expressed as followed:
d ..
(d
)
t dt(A 2V,A 2V)H =2t dt v,Asv H
= - 2t(Au, ASV)H + 2t(FI( v + l'](B t w)), ASV)H
+ 2t(F2(v + rJ(Btw)) ,ASV)H. We have
(Av , ASV)H
= IIA~+ ~vIIH = Ilvlli+s'
Similar to the argument of [21] and th e estim ate for the existence of absorbing, and applying some embedding theorems, see Temam [18] Page 12 we have got
Ilvll; ::; C(t , IlvollH , sup tElO,I]
111']1IlD(A')) ' for t
E [0,1] .
72
VENA PEARL BONGOLAN-WALSH ET AL .
By the results of [12] and [21], we know sup 11711 IlD(AO) ::; C(trL2Q) <
00,
tElo . I}
for
0<s<
1
4'
By now, our est imates allow us to write down th e main assertio n with respect to the dissipati vit y of thi s section. THEOREM 4 .3 . For the random dynamical syst em generated by [ l S}, there exists a compact random set B = {B(W)}WEfl which satisfies Definition 4.1. We define
(22)
0<s<
1
4'
In particular, 1{s is compactly embedded in H . 5. Random dynamics: Enstrophy and ergodicity. In this section we will apply the dissipativity result of the last section to analyse the dyn amical behavior of the random dynamical system (1). However, it will be enough to analyse the transformed random dynamical system generated by (13). By the transformation (12), we have all these qualitative properties to the origin al gravity current system (1). We will consider the following dynamical behavior : rand om at t ractors, asymptot ic evolution of enst rophy and mean-square norm of salinity profile, and ergodicity. Enstrophy is defined as one half of th e mean-squ are int egral of vorticity. Ergodicity implies t hat the time average for observables of the dynamical syst em approximates th e statistical ensemble average , as long as t he time inte rval is sufficiently long. We first consider rand om attractors. We recall t he following basic concept; see, for instan ce, Fland oli and SchmalfuB [9]. DEFINITION 5.1. Let sp be a random dynam ical system. A random set A = {A(W)} WEfl consisting of compact non empty sets A(w) is called random global attractor if for any random bounded set D we have for the limit in probability
(JPl) lim distH( cp(t ,w, D(w)) , A(Otw)) = 0 t-+ oo
and cp(t, w, A(w) ) = A (Otw) any t 2': 0 and wEn . The essential long-time behavior of a random system is captured by a random attractor. In the last section we showed that the dynamical system sp generated by (6) is dissipative which means that there exists a random set B satisfying (21). In addition, this set is compact . We now recall and ad ap t the following theorem from [9].
ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS
73
THEOREM 5.2. Let cp be a random dynamical dynamical system on the state space H which is a separable Banach space such that x ----l cp(t,w,x) is continuous. Suppose that B is a set ensuring the dissipativity given in Definition 4.1. addition, B has a subexponential growth (see Definition 4.1) and is regular (compact) . Then the dynamical system cp has a random
In
aitractor. This theorem can be applied to the random dynamical system cp generated by the stochastic differential equation (13). Indeed, all the assumptions are satisfied. The set B is defined in Theorem 4.3. Its subexponential growth follows from B(w) c B(O, R(w)) where the radius R(w) has been introduced in the last section. Note that cp is a continuous random dynamical system; see Theorem 3.lt Thus cp has a random attractor. By the transformation (12), this is also true for the original gravity currents system (6). COROLLARY 5.3 (Random Attractor). The gravity currents system (6) has a random attracior. Now we consider random fixed point (stationary state) and ergodicity. DEFINITION 5.4 . A random variable v* : n ----l H is defined to be a random fixed point for a random dynamical system if
cp(t,w,v*(w)) = v*(Btw)
°
for t ~ and wEn . A random fixed point v* is called exponentially attracting if
lim Ilcp(t ,w,x) - v*(BtW)/IH =
t-.oo
°
Sufficient conditions for the existence of random fixed points are given in Schmalfuf [14] . We here formulate a simpler version of this theorem and it is appropriate for our system here. THEOREM 5.5 (Random Fixed Point Theorem). Let cp be a ranfor any x E Hand wEn .
dom dynamical system and suppose that B is a forward invariant complete set . In addition, B has a subexponential growth , see Definition 4.1. Suppose that the following contraction conditions holds :
(23)
sup
/Icp(1,w ,vI)-cp(1,w,v2)IIH
where th e expectation oflog k denoted by lE log k < O. Then cp has a unique random fixed point in B which is exponentially attracting.
This theorem can be considered as a random version of the Banach fixed point theorem. The contraction condition is formulated in the mean for the right hand side of (23) . THEOREM 5.6 (Unique Random Stationary State) . Assume that the salinity boundary flux data 11F11L 2 , the Prandtl number Pr, and the trace of the eovariance for the noise trL 2Q are sufficiently small. Then
74
VENA PEARL BONGOLAN-WALSH ET AL .
the random dynamical system generated by (6) has a unique exponentially attracting random stationary state.
Note that if we take into account the horizontal and vertical diffusivities K.H, K.v in the Laplacian operator, the above "smallness" condit ion needs to be slightly modified. Here we only give a short sketch of the proof. Let us suppose for a while that B is given by the ball B(O, R) introduced in Lemma 4.2. Suppose that the data in the assumption of the lemma are small and v is large. Then it follows that !ER is also small. To calculate the contraction condition we have to calculate 1I
o.
THEOREM 3.1. Assume the hypothesis (HI) ' Then there exists a unique invariant probability 7f for the Markov process X n := an...aIXO, where Xo is indepen dent of {an := n ~ I}. Also, one has
(3.11)
d(T*np"
7f)
:S (1 - 0)[nlNI
(p,€P(S))
where T*n is the distribution of X n when X o has distribution J.L, and [n/N] is the integer part of nfN;
RANDOM DYNAMICAL SYSTEMS IN ECONOMICS
189
We now state two corollaries of Theorem 3.1 applied to LLd. monotone maps. Corollary 3.1 extends a result of Dubins and Freedman, [1966 Thm. (5.10)], to more general state spaces in ~ and relaxes the requirement of continuity of an ' The set of monotone maps may include both nondecreasing and nonincreasing ones. Let S be a closed subset or an interval of~. Denote by dK(J.l, v) the Kolmogorov distance on P(S) . That is, if Fp. , F; denote the distribution functions (d.f) of J.l and i/, then
dK(J.l, v) = sup 1Fp.(x) - Fv(x)1 (3.12)
X €s
== sup 1Fp.(x) - Fv(x)l , (J.l , VEP(S)). X€!R
It should be noted that convergence in the distance d« on P(S) implies weak convergence in P(S). COROLLARY 3.1. Let S be an interval or a closed subset of~. Suppose an(n ~ 1) is a sequence of i.i.d. monotone maps on S satisfying the splitting condition (H) : (H) There exist XOES, a positive integer Nand a constant fJ > 0 such that
Prob(aNaN-I aIX Prob (aNaN-I aIx
~
~
Xo \fxES) Xo \fxES)
~
fJ
~ fJ
(aJ Then the sequence of distributions T*nJ.l of X n := an...aIXO converges to a probability measure 7l' on S exponentially fast in the Kolmogorov distance dk irrespective of X o. Indeed, (3.13)
where [y] denotes the integer part of y. (bJ tt in (aJ is the unique invariant probability of the Markov process X n . Proofs of Theorem 3.1 and Corollary 3.1 are spelled out in Bhattacharya and Majumdar [3, 4]. 3.2. Applications of splitting. 3.2.1. Stochastic turnpike theorems. We now turn to the problem of economic growth under uncertainty. A complete list of references to the literature - influenced by ther works of Brock and Mirman - is in Majumdar, Mitra and Nyarko [8]. I indicate how the principal results of this literature can be obtained by using Corollary 3.1. Instead of a single law of motion (1.9), we allow for a class of admissible laws with properties suggested by the deterministic Solow model in its reduced form [see (1.10) and (1.11)]. Consider the case where S = ~+; and r = {FI , F2 , .. . , Fe, ..., FN } where the distinct laws of motion F; satisfy:
190
MUKUL MAJUMDAR
(F.1) Pi is strictly increasing, continuous, and there is some r, > 0 such that Fi(x) > x on (0, r i) and Fi(x) < x fOT X > ri. Note that Fi(ri) = Ti for all i = 1, ..., N. Next , assume: (F.2) r, =I rj for i =I j. In other words , the unique positive fixed points r i of distinct laws of motion are all distinct. We choose the indices i = 1,2, ..., N so that
Let Prob (an = Fi) = Pi > O(i ::; i ::; N) . Consider the Markov process {Xn(x)} with the state space (0,00). Ify ~ rI , then Fi(y) ~ Fi(rl) > rl for i = 2, ...N , and F1(TI) = rI, so that Xn(x) ~ TI for all n ~ 0 if x ~ TI. Similarly, if y ::; TN, then F;(y) ::; F;(rN) < TN for i = 1, ..., N - 1 and FN(TN) = r», so that Xn(x) ::; rN for all n ~ 0 if x ::; rn . Hence, if the initial state x is in [rl,rN], then the process {Xn(x) : n ~ O} remains in h, TN] forever. We shall presently see that for a long run analysis we can consider h ,r N] as the effective state space. We shall first indicate that on the state space h , r N] the splitting 2)(x) ::; F1(x) etc. The condition (H) is satisfied. If x ~ rI, F1(x) ::; x, Fi limit of this decreasing sequence F1(n) (x) must be a fixed point of F1 , and therefore must be rl. Similarly, if x ::; rN , then FfJ(x) increases to r N. In particular,
Thus, there must be a positive integer no such that
Fino)(rN) < F~nO)(rl)' This means that if Zo E
[pino) (r N), Fino) (rd]' then
Prob(Xno(x) ::; Zo 'v'xEh,rN]) ~ Probfo., = F 1 for 1 ::; n ::; no) =
P~o
>0
Prob(Xno(x) ~ Zo 'v'xEh, Tn]) ~ Probfo., = FN for 1 ::; n ::; no) = p';Y > O. Hence , considering h, r N] as the state space , and using Theorem 3.1, there is a unique invariant probability 7r with the stability property holding for all initial XEh , rN] ' Now, define m(x) = . min Fi(x) , and fix the initial .=I ,...,N
state XE(O, rr). One can verify that (i) m is continuous; (ii) m is strictly increasing; (iii) m(rI) = rI and m(x) > x for XE(O, TI), and m(x) < x for x > rI. Clearly m(n)(x) increases with n, and m(n)(x) ::; rI. The limit of the sequence m(n)(x) must be a fixed point, and is, therefore "i- Since Fi(rI) > rI for
RANDOM DYNAMICAL SYSTEMS IN ECONOMICS
191
i = 2, ..., N , there exists some e > 0 such that Fi(y) > rl (2 :S i :S N) for all YE[rl - c,rI] . Clearly there is some ne such that mn«x) ~ rl - c. If 71 = inf{n ~ 1 : Xn(x) > ri} then it follows that for all k ~ 1
Prcbf r,
> ne + k) :S p~ .
pt
Since goes to zero as k ~ 00 , it follows that 71 is finite almost surely. Also, X T 1 (X) :S rN, since for y:S rI, (i) Fi(y) < Fi(rN) for all i and (ii) Fi(rN) < rn for i = 1,2, ...,N -1 and FN(rN) = rN. (In a single period it is not possible to go from a state less than rl to one larger than rN) . By the strong Markov property, and our earlier result , XT+m( x) converges in distribution to IT as m ~ 00 for all u(O ,rI) . Similarly, one can check that as n ~ 00, Xn( x) converges in distribution to IT for all x > rN. The assumption that I' is finite can be dispensed with if one has additional structures in the model. Here is a simple example.
3.2.2. Uncountable I': an example. Let F: R+ ~ R+ satisfy: (F.1) F is strictly increasing and continuous. We shall keep F fixed. Consider e = [B 1 , B2 ], where 0 < B1 < B2 , and assume the following concavity and "end point" conditions: (F.2) F(x)/x is strictly decreasing in x> 0, 02~~:;") < 1 for some x" > 0
01F~x') > 1 for some x' > O. Since BF(x) is also strictly decreasing x x in x , F .1 and F.2 implies that for each BEe, there is a unique Xo
>0
BF(xo) . BF(x) such that = 1, i.e., BF(xo) = x «. Observe that - - > 1 for Xo x BF(x) 1£ N B' B'" u O < x < Xo , - - < or x > Xo. ow, > impues Xo' > xo" ; x ()'F(xo") ()"F(xo") ()' F(xo') W. --'--:"'-;" > = 1= Xo' > XO". nte Xo" Xo" Xo'
= U : f = BF, BEe}, I, = ()I F I'
12 =
and
()2F.
e
Assume that B is chosen LLd. according to a density function g(B) on which is positive and continuous on e . In our notation, h (xo1) = xo1; 12(xoJ = xo2 • If x ~ XO\l f(x) == BF(x) ~ f(xoJ ~ h(xo 1) = xo1. Hence Xn(x) ~ x01 for all n ~ 0 if x ~ xo1. If x :S X0 2
Hence , if x E [xo 1, xo2] then the process X n (x) remains in [xo 1, xo2 ] forn)(xo = x0 and lim f~n)(X01) = xo . There must ever . Now, lim fi 2 2) 1 n ~oo
n -lo OO
be a positive integer no such that fino) (xo2 ) < f~no) (xo1). Choose some
192
MUKUL MAJUMDAR
no) (X0 ) , fino)(xo ) ). There exist intervals [rh, 0 + m], [0 -m', O 2 1 2] l 2 such that for all 0 € [01,01 + m] and BE[02 - m , O2 ] Zo € ui
Then the splitting condition holds. Now fix x such that 0
< x < XO I , then
Let m be any given positive integer. Since (OIF)(n)(x) ---+ XO I as n ---+ 00, 1 there exists n' == n'(x) such that (OIF)(n)(x) > XO I - - for all n > n' . This m = 1 implies that Xn(x) > XO I - - for all n 2 n'. Therefore, lim infXn(x) 2 m XOI ' We now argue that with probability one, lim infXn(x) > XOI ' For n~~
.
this, note that
. If we choose
n-+oo
~-~
0 = --2- and e
> 0 such that XO
I
-
e>0
then min{OF(y) - fhF(y) : 0 2 (h + 0, y 2 XO I - e} 2 of(xol - e) > O. Write 0' == of(xo l - e) > O. Since with probability one, the LLd. sequence {e(n) : n = 1,2, ...} takes values in [e l + 0, e2 ] infinitely often so that 1 lim infXn(x) > xO I - 1.. + 0'. Choose m so that - < 0'. Then with n ~ oo m m probability one the sequence Xn(x) exceeds XOI' Since x02 = fJn)(xo2 ) 2 X n(xo2 ) 2 Xn(x) for all n, it follows that with probability one, Xn(x) reaches the interval [xo I , xo2 ] and remains in it thereafter. Similarly, one can prove that if x > X0 2 then with probability one, the Markov process Xn( x) will reach [xo l l x o2 ] in finite time and stay in the interval thereafter. REMARK 3.1. The proof of this result holds for any bounded nondegenerate distributioin of e (if E is the support of the distribution of e, define el == inf c < e2 == sup £). 3.2.3. An estimation problem. Consider a Markov chain X n with a unique stationary distribution 71". Some of the celebrated results on the ergodicity and the strong law of large numbers hold for rr-almost every initial condition. However, even with [0,1] as the state space the invariant distribution 71" may be hard to compute explicitly when the laws of motion are allowed to be non-linear, and its support may be difficult to determine or may be a set of zero Lebesgue measure [see Bhattacharya and Rao [2]] . Moreover, in many economic models, the initial condition may be historically given, and there may be little justification in assuming that it belongs to the support of 71". Consider then a random dynamical system with state space [e, dJ (without loss of generality for what follows choose e > 0). Assume r consists of a family of monotone maps from S with S, and the splitting condition (H ) holds. The process starts with a given x. There is, by Corollary 3.1, a unique invariant distribution (a stochastic equilibrium) 71" of the random dynamical system, and (3.13) holds. Suppose
RANDOM DYNAMICAL SYSTEMS IN ECONOMICS
we want to estimate the equilibrium mean ~
n-1
L: X j .
Is Y7f'(dy) by
193
sample means
We say that the estimator ~ L: X, is vn-consistent if
j=O
j=O n-1
~ L x, =
(3.14)
j=O
J
Y7f'(dy)
+ Op(n- 1/ 2 )
where Op(n- 1/ 2 ) is a random sequence en such that len/n- 1/ 21 is bounded in probability. Thus, if the estimator is vn-consistent, the fluctuations of the empirical (or sample-) mean around the equilibrium mean is Op(n- 1/ 2 ) . We shall outline the main steps in the verification (3.14) in our context. For any bounded (Borel) measurable f on rc, dj, define the transition operator T as:
Tf(x) = hf(y)p(x,dY) By using the estimate (3.13), one can show that (see Bhattacharya and Majumdar [4]' pp. 217-219) if f(z) = z - I Y7f'(dy) then 00
00
sup
L
x
n=m+1
ITnf(x)1 ~ (d - c)
L
(1 - 8)[n/N]
--->
0
as m
---> 00
n=m+1 00
Hence, 9 = -
L: T N f
[where TO is the identity operator I] is well-defined,
n=O 00
and g, and Tg are bounded functions. Also, (T - 1)g
= - L: T" f + n=l
00
L: TN f = f . Hence, n=O n-1
n-1
L f(Xj) = L(T - I)g(Xj) j=O
j=O n-1
= L((Tg)(Xj) - g(Xj)) j=O n
= L[(Tg)(Xj-d - g(Xj)] + g(Xn )
-
g(Xo)
j=l
By the Markov property and the definition of Tg it follows that
where Fr is the a-field generated by {Xj : 0 ~ j ::s; r} . Hence, (Tg)(Xj_I)g(Xj)(j ~ 1) is a martingale difference sequence, and are uncorrelated,
194
MUKUL MAJUMDAR
so that n
k
(3.15)
EL [(Tg(Xj-d - g(X j ))]2 = LE((Tg)(Xj-d - g(X j ))2. j=1
j=1
Given the boundedness of 9 and Tg, the right side is bounded by n .a for some constant a . It follows that for all n where r/ is a constant that does not depend on X o. Thus, n-1
E(~ ~ x, J=O
J
Y1r(dy))2 ::; r/ln
which implies, n-1
~ L x, = j=O
J
Y1r(dy)
+ Op(n- 1/ 2).
For other examples of vn-consistent estimation, see Bhattacharya and Majumdar [4] . 4. Iterates of quadratic maps. On the state space S = (0,1) consider the Markov process defined recursively by
X n +1 = a g n + 1 X n (n = 0, 1,2, ...)
(4.1)
where {en : n ~ 1} is a sequence of LLd. random variables with values in (0,4) and , for each value (}£(0,4), a() is the quadratic function (on S):
(4.2)
aox
== ao(x) = Ox(l
- x)
o<x<1.
As always, the initial random variable X o is independent of {en : n ~ 1}. Our main result from Bhattacharya and Majumdar [5] provides a criterion for Harris recurrence and the existence of a unique invariant probability for the process {Xn : n ~ O}. Recall that a sequence J.Ln(n ~ 1) of probability measures on S is said to be tight if, for every e > 0, there exists a compact K g C S such that J.Ln(K g) ~ 1 - e for all n ~ 1. THEOREM 4 .1. Assume that the distribution of e1 has a nonzero absolutely continuous component (w.r .t . Lebesgue measure on (0,4) whose density is bounded away from zero on some nondegenerate interval in (1,4). If, in addition, E~=1P(n)(x ,dy) : N ~ I} is tight on S = (0,1) for some x, then (i) {Xn : n ~ O} is Hams recurrent and has a un ique invariant
{*"
RANDOM DYNAMICAL SYSTEMS IN ECONOMICS
195
-k
probability 1T" and (ii) 'L,:=1 p(n)(x, dy) converges to 1T" in total variation distance, for every x, as n ~ 00 . COROLLARY 4.1. If Cl has a nonzero density component which is bounded away from zero on some nondegenerate interval contained in (1,4) and if, in addition,
(4.3)
E log
Cl
>
°
and E Ilog(4 -
cd/ < 00,
then {Xn : n 2: O} has a unique invariant probability 1T" on S = (0,1) and (l/N) 'L,:=1 p(n)(x, dy) ~ 1T" in total variation distance, for every X€(o, 1). REMARK 4.1. Under the hypothesis of Theorem 4 .1, the Markov process is not in general aperiodic. For example, one may take the distribution of En to be concentrated in an interval such that for every () in this interval no has a stable periodic orbit of period m > 1. One may find an interval of this kind so that the process is irreducible and cyclical of period m . If Cn has a density component bounded away from zero on a nondegenerate interval B containing a stable fixed point, i.e., Bn(O, 3) -I cP, then the process is aperiodic and p(n)(x, .) converges in total variation distance to a unique invariant 1T". Assumptions of this kind have been used by Bhattacharya and Rao [2] and Dai [6] .
REFERENCES [1] BHATTACHARYA RN. and E.C . WAYMIRE: Stochastic Processes with Applications, John WHey, New York (1990).
[2] BHATTACHARYA R.N . and B.V. RAO: Random Iterations of Two Quadratic Maps .
[3] [4] [5J [6] [7]
[8] [9] [10] [11]
In Stochastic Processes (eds. S. Cambanis, J.K Ghosh, RL. Karandikar, and P.K Sen), Springer Verlag, New York (1993), pp. 13-21. BHATTACHARYA R.N. and M. MAJUMDAR: On a Theorem of Dubins and Freedman , Journal of Theoretical Probability, 12 (1999), pp. 1067-1087. BHATTACHARYA RN . and M. MAJUMDAR: On a Class of Stable Random Dynamical Systems: Theory and Applications, Journal of Economic Theory, 96 (2001) , pp. 208-229. BHATTACHARYA R.N. and M. MAJUMDAR: Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes, CAE Working Paper 02-03, Cornell University, (2002) [to appear in Annals of Applied Probability.] DAI J .J .: A Result Regarding Convergence of Random Logistic Maps , Statistics and Probability Letters, 47 (2000), pp. 11-14. DUBINS L.E. and D. FREEDMAN (1966): Invariant Probabilities for Certain Markov Processes, Annals of Mathematical Statistics, 37, pp. 837-858. MAJUMDAR M., MITRA T ., and Y. NYARKo: Dynamic Optimization under Uncertainty; Non-convex Feasible Set . In Joan Robinson and Modern Economic Theory (ed. C.R. Feiwel), MacMillan, London (1989), pp. 545-590. MAJUMDAR M. and T . MITRA: Robust Ergodic Chaos in Discounted Dynamic Optimization Models, Economic Theory, 4 (1994), pp. 677-688. MITRA K : On Capital Accumulation Paths in a Neoclassical Stochastic Growth Model, Economic Theory, 11 (1998), pp. 457-464. SOLOW R.M.: A Contribution of the Theory of Economic Growth, Quarterly Journal of Economics, 70 (1956), pp. 65-94.
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION OF THE NAVIER-STOKES EQUATIONS M. ROMITO· Abstract . We explain some ideas contained in some recent pa pe rs, concerni ng the st at ist ical long time behaviour of the spectral approximation of the Navier-Stokes equations, driven by a highly degenerate white noise forcing. The analysis highlights th at the ergodicity of the stochastic system is obtained by a geometric cascad e. Such a cascade can be int erp reted as the mathematical counterpart of th e energy cascade , a well-known phenomenon in turbulence. In th e second pa rt of the paper, we analyse the results of some numerical simulations. Such simulations give a hint on the beh aviour of the system in the case where th e whit e noise forcing fails the assumptions of the main theorem . Key words. Navier-Stokes equations, ergodicity, hypo-ellipticity, cont rollability. AMS(MOS) subject classifications. 76M35, 76F55 .
Primary 76D05;
Secondary 35Q30,
1. Introduction. The article aims to explain a few ideas related to the analysis of the statistical long time behaviour of a model of isotropic homogeneous turbulence, that have been presented in some recent papers. We will refer mainly to Romito [33], of whom we will present the approach, but also to E and Mattingly [11], who originally solved the problem in the 2D setting. Moreover, we will also refer to Agrachev and Sarychev [1], for the section on the irreducibility property (Section 5). The model equations are the 3D Navier-Stokes equations in a cube , with periodic boundary conditions, driven by a white noise random force. Due to the well-known analytical difficulties related to the problem , we will examine the spectral approximation of the equation.' Namely, we will consider the Fourier series of the solution up to a fixed, arbitrary but finite, threshold. The approach has anyway a qualitative interest for the general problem. Indeed , according to Kolmogorov's theory of turbulence (Kolmogorov [21]) , the cascade of energy, responsible of the transport of the energy through the scales, is mainly effective only in the inertial range , and it becomes negligible at smaller scales, where the dissipation ends up to be the only relevant phenomenon. Hence the long-time statistical properties of the fluid can be sufficiently depicted by the low modes of the velocity field. In some sense, if the ultraviolet cut-off threshold is sufficiently large , in order to capture all the important modes, the corresponding invariant measure, • Dipartimento di Matematica, U niversita di Firenze, Viale Morgagni 67/a, 50134 Fi renze , Italia (romito~ath.unifi.it). 1 In a recent paper by Da Prato and Debussche [6] , a proof of ergodicity is given for the infinite dimensional stochastic PDE. We just remark that, among other assumptions, the noise acts on all Fouri er modes . 197
198
M. ROMITO
E(k)
[ne~ange
~
Dissipation
----, ~_-~~
r
~
injection of energy by large-scale
noise forcing
forcing
•• •• • ~~'lY ca.<>lde
. . .... .. . ~ Ikl
FIG . 1.
Th e energy spectrum .
which is the mathematical object representing the asymptotic behaviour, should give t he real picture of the dynamics of the fluid (the analysis we have described t urns out to be true in a purely deterministic setting, as it has been proved by Constantin, Foias and Temam [4]). A very rough picture of the Kolmogorov spectru m is given in Figure 1. Energy is injected by a deterministic forcing in th e system at the largest scales. The rough forcing injects energy in th e system at scales much smaller but larger than the dissipation scale. One should imagine th at the deterministic components correspond to forcings like gravity, while th e stochastic forcing corresponds to small scales forcings like mechanical vibrations, or temperature fluctuations , etc. In our model the noise injects energy at the level of the largest lengthscales, since we neglect the deterministic forcing. It means that the noise acts only on a few of the total number of modes taken into account, up to the spectral threshold, for the description of the syst em. In principle , one could expect that not all of the component of the system are dumped by the noise. The main point of the paper is that indeed all components of the system are, either directly or indirectly, dumped by th e noise. We call this phenomenon the geometric cascade, because we will trace the noise dumping, component by component, from the forced Fourier modes to all the ot hers (this is the heurist ic interpretation of Lemma 4.1). E and Mattingly [11] showed, in t he 2D case, that this mechanism proves effective to deduce that the transition probabi lity densities of the Markov process , solut ion to the model equation, are regular. In Romito [33], it is proved (see also Theorem 5.1) that the geometric cascade is effective also in showing that the process is steered by the noise to any part of the state space , with positive probability; such a property is known as irreducibility. A toy model, that is, a very simple SDE, is given in Section 6, to show how the concepts and methods explained in the article apply practically. An open question related to the analysis above concerns the sit uation in the non hypo-elliptic case. Assume that the noise does not excite
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 199
(directly or indirectly) all the modes. Heuristically, the system should experience a decoupling between the forced modes and the modes not excited by the noise. On one hand, the modes forced should behave as in the hypoelliptic situation, hence converging to a statistical steady state, which will be necessarily concentrated on a hyper-plane. On the other hand, the nonforced modes should behave as in the case of absence of noise, converging to zero. In the last part of the paper (see Section 7), we try to obtain a hint on this heuristic picture from the analysis of some numerical simulations. The aim is to verify that some modes, the ones that supposedly are not forced, either directly or indirectly, by the noise, converge to zero. Our analysis aims to give just a qualitative answer to our conjecture, without any pretension of a precise quantitative analysis. 2. The Navier-Stokes equations in the Galerkin approximation. Consider the stochastic Navier-Stokes equations (2.1)
du, = (v.6.u - (u· V')u - V'p) dt + dB t , divu = 0
in the domain x E [0, 27f]d, where d = 2,3, with periodic boundary conditions. Following the classic approach, that dates back to the pioneering works of Leray (see [23]' [24], [25]), we will consider the equations projected onto the space of divergence-free vector fields. Such a projection reads easily in our case, namely of periodic boundary condit ions, since it is diagonal with respect to the Fourier basis (e1k.x)kE71d . Indeed , the Helmholtz-Leray projection is defined as (2.2)
where &k is the projection in IR d onto the orthogonal space to the vector k and it is given by d= 2, d= 3,
where kl. = (-k 2 , kI) is the vector perpendicular to k. If we write the solution u(t,x) in the Fourier basis, namely
u(t,x) =
L
uk(t)e 1k'x ,
kE71d
the deterministic dynamics is given by (2.3) 9"(v.6.u - (u · 'V )u - 'Vp)
=-
L (v 1k I + L 2
kE71 d
i
(k · Uh) 9"k(ul))eik'X,
h+l=k
200
M. ROMITO
the pressure term being disappeared because of the projection onto the divergence free vector space . We will assume also that Uo == 0, that means that we observe the system in a reference frame centred on the centre of mass of the fluid. There is no loss of generality, since the forcing has zero average, hence the centre of mass moves with constant velocity. 2.1. The noisy forcing term. As it has been remarked in the introduction, the forcing term is a white noise, the time derivative of a Brownian motion. It is the simplest, yet quite significant, model for rough and unpredictable forcing. We will assume that the covariance of the noise is diagonal with respect to the Fourier basis , so that
e, = L
(2.4)
qk . j3fei k .x
kEZ d
where (j3fh~o, k E tl d are independent d-dimensional Brownian motions and the qk are d x d matrices such that q~ . k = O. The last condition means that the Brownian motion has trajectories which are divergence free. We will assume that most of the qk are zero, so that we define the set of forced modes as
N
(2.5)
= { k E tl
d
Iqk ~ 0 } .
For the sake of simplicity, we will assume that for each kEN, the matrix qk has rank d - 1. 2.2. The Galerkin approximation. The Navier-Stokes equations, specially in three dimensions, are difficult to be managed. Hence, we will consider a spectral approximation of the equation. Let N E IN be an integer, and consider the truncated representation UN (t, x)
=
L
Uk(t)e
ik x .
Ikloo:$N
where JkJ oo = max(lkd, · . . , Ikdl) . The equations for the truncated system can be derived by projecting the representation for the deterministic dynamics (2.3), so that at the end 7Lk=-[vlkI2+i (2.6)
:L (k 'Uh).9'k(uI)]dt+qk dj3f , h+l=k
Uk '
k
= 0,
with Ikloo ::; N , and the sum is extended to all hand 1 smaller than the threshold N.
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 201
3. The ergodic properties of the process. The system of stochastic differential equations (2.6) defines a Markov process on the state space
where D = 2d[(2N + l)d -1] and we understand that Uk = rk generator of the Markov process is given by (3.2)
2" = F
+~
L (Xk:
2
+ iSk. The
+ Xk2 )
kEN
where F = Llkl~N Fk: a~k +Fka~k is the vector field given by the deterministic dynamics and Xk: , Xk are the vector fields given by the noise forcing. Finally, we will denote by (Pd t20 the Markov semigroup associated to the Markov process U = (uk)lkl~N' The main result we want to show says that, under suitable assumptions on the number of forced modes (see Definition 4.1), the Markov process is ergodic: it has a unique invariant measure and the distributions of th e process converge exponentially to the unique invariant measure. The main tool is the following theorem. THEOREM 3.1 (Doob [8]) . Given a Marko v semigroup (Pdt2 0 and an invariant measure f.l for the semigroup, if there is a time to such that the probability measures Pt(u, ·) are mutually equivalent for all t ;::: to and u E U, then f.l is the unique invariant measure, it is equivalent to all Pt(u, ') , for t ;::: to, and it is strongly mixing. The assumptions of Doob 's theorem can be fulfilled under the assumptions of the following theorem, due to Has'minskif. THEOREM 3.2 (Has'minskii [16]). If the Markov semigroup (Pdt2 0 is strongly Feller and irreducible, then all probability m easures Pt (u , .) are mutually equivalent. The previous result is the key point for the proof of the main theorem of the paper. The claim on the support of the invariant measure is a direct consequence of the irreducibility (see Stroock and Varadhan [36]) . THEOREM 3.3. A ssume that the set of indices N corresponding to the forced modes is a determining set of indices (as defined in Definition 4.1) . Then the Markov process solution to the stochastic differential system (2.6) is ergodic. Moreover, the unique invariant measure is fully supported on the state space U . We will see in the next section the precise definition of a determining set of indices. In the two dimensional case, some examples are the set of modes {(I , 0), (1, I)} (see E and Mattingly [11]) or the set of modes
{(M
+ 1,0), (M, 0), (0, K + 1) , (0 , K)} ,
for suitable integers M and K (see Mattingly and Pardoux [27]) . In the three dimensional case, the set of modes
202
M. ROMITO
(1,0,0),(0,1 ,0),(0,0,1) is given in Romito [33] . REMARK 3.1. It can be proved that the convergence stated in the previous theorem is indeed exponentially fast (see Romito [33J) . Such a claim follows by applying Theorem 6.1 of Meyn and Tweedie [31]. The key point is to show that there is a Lyapunov function, that is, a positive function V, defined on the state space, such that
if V
:s; -cV + d
°
for some constants c > and d. A Lyapunov junction for the problem under examination is the kinetic energy 2:k IUkI 2 . 4. Regularity of the transition probabilities. The first main point of the proof of Theorem 3.3 is to prove that the transition semigroup has the strong Feller property. It means that for each bounded measurable function ep defined on the state space U, the function Ptep is bounded continuous. First assume that the noise excites all modes. In our notations, K contains all indices Ikl oo :s; N or, equivalently, the covariance f5j of the noise is a non-singular matrix. The Bismut-Elworthy-Li formula (see Bismut [2] and Elworthy and Li [10]) gives
(D(Pt'P)(UO),V)
=
1 -E['P(u(t,uo» t
it 0
(..P-1Du(s,uo)v,dB s ) ] ,
where u(t, uo) is the process , solution to (2.6), starting at Uo (one can see also Cerrai [3]). It turns out that P; has a regularising effect, in particular it implies the strong Feller property. In the general case, where K contains just a few modes.f one needs to have better information on the non-linear dynamics, which is ultimately responsible of the spreading of the noise forcing to all modes. Here we need the probabilistic version of Hormander celebrated theorem. THEOREM 4.1 (Malliavin [26], Stroock [35]). Consider the Stratonovich SDE in IRffi n
«x, =
F(Xt ) dt +
L Fi(X
t)
0
dW;,
i=l
Xo=x, 2 As it has been proved in E and Mattingly [11], for the spectral approximation of 2D Navier-Stokes equations, two modes, namely (1,0) and (1,1) are enough. In the 3D case, as it can be found in Romito [33], the three modes (1,0,0), (0,1,0), (0,0,1) are sufficient .
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 203
where the vector fields F , F i , . . . , Fn satisfy suitable boundedness conditions, and assume that Hormander's condition holds: the Lie algebra generated by the vector fields F , F i , ... , F n , once evaluat ed at x , spans IR m . Then, for all t > 0, the random variable X, has an absolutely continuous distribution with a sm ooth COO density. Such a result has been established by Malliavin [26], and in its full strength by Stroock [34], [35]. A simplified argument is given in Norris [32]. In order to analyse how the non-linearity spreads the effect of the noise, we notice that the state space U, defined in (3.1), can be written as a direct sum of linear subspaces, namely,
where each point of Uk has all coordinates equal to zero , but the coordinates Uk corresponding to k, and k . Uk = O. In the same way we can define the Lie algebra U of vector fields 9 = L Gk a~k such that k . Gk = O. Moreover, we will consider the subspaces Uk of U of constant vector fields whose coefficients are in Uk . Now the main technical lemma follows. LEMMA 4.1. Let m and n be indices, such that Iml oo , [n] 00 ~ N , and consider two vector fields V E Urn , WE Un' Then (i) if m 1\ n = 0, then [[F,V], W] = 0; (ii) [[F,V] , W] = ~[[F, V + W], V + W] . Moreover, if m 1\ n i= 0, [m], i= Inl2 and [m + nl oo ~ N ,3 then the Li e algebra Urn +n is contained in the Lie algebra generated by the vecto r fields [[F, V], W], with V E Urn and W E Un . The last part of the lemma is the main point. We can interpret it in the following way: if the modes m and n are forced by the noise, the mod es corresponding to m + n are in the Lie algebra generated by the vector fields F and Xk. It means that the (m + n)-component of the system of SDEs is forced indirectly by the noise. Those components, again, can transmit the noise force to other components, and so on. If the set N of directly forced mod es generates the set of all modes, the equ ations behave as if some kind of forcing happens at each component. This is truly a geometric cascade for the Fourier modes of the Navier-Stokes equations. In order to substantiate the idea, we introduce the set A(N) with the following rules:
Ne A(N) , if k E A(N) , -k E A(N) , [> if m , nE A(N), m 1\ n i= 0, Iml2 i= Inl2 and [m + nl oo ~ N , then m+n E A(N). DEFINITION 4.1 (Determining sets of indices) . We say that a set [> [>
le c {k Ilkl oo ~ N} is a determining set of indices for the threshold N if the set A(N) contains all indices Ikloo ~ N . 3Here
Irnl2
is the Euclidean norm of the vector
rn,
namely Irnl ~ =
L rn;.
204
M. ROMITO
Consequently, we have the following result. THEOREM 4.2 . If the set N is a determining set of indices, then the Markov semigroup associated to the SDE (2.6) has the strong Feller property. 5. Irreducibility and the control problem. A Markov process is said to be irreducible if for each time T and each open set A of the state space, the probability that the process is in A at time T is positive. The irreducibility property is implied (see for example Stroock and Varadhan [36]) by a controllability property of the control problem which is obtained from the original system (2.6) by replacing the noise with some controls . In our case, we obtain the system
where Pk is a quadratic polynomial in the Uk and Vk are the controls. We say that the system is controllable if for each time T and each pair of points 1 2 U , u , there exists a control v such that the corresponding solution of the control problem starts at time 0 in u 1 and stops at time T in u 2 • Intuitively, the irreducibility is related to the control problem because, if the system is controllable, then this means that there are realizations of the noise such that the Markov process solution of the SDE can move from u 1 to u 2 . A necessary condition for such a property is indeed the hypo-ellipticity (see Jurdjevic [18] for generalities on the geometric control theory) . Heuristically, if at each point of the state space , our system is allowed to follow any direction in any way (only the first assumption is true in the hypoellipticity proof) , then the system is controllable. Roughly speaking, the Lie brackets that have been evaluated in the previous section represents all the possible directions that the process can follows . Indeed, the drift of the equations would lead the process to follow the field :F of the deterministic dynamics . The kicks of the noise allow the process to follow other directions, which are a combination of the original dynamics :F and the directions Xk of the kicks. If the polynomial P has odd degree, the hypo-ellipticity condition is also sufficient (see Jurdjevic and Kupka [19]), since by changing u ----+ -u , the system can always follows both ways of each direction. The problem is, the Navier-Stokes system gives rise to a polynomial of even degree. Such quadratic terms in some sense give a preferential direction to be followed by the system. In order to give a clearer idea of such phenomena, one can see the following section , where an extremely simple SDE is presented, which gives a hypo-elliptic diffusion, and such that the upwards direction is preferential. The key property that the spectral approximation of the Navier-Stokes equations enjoys is the absence of such dangerous quadratic terms, and it is a consequence of properties (i) and (ii) of Lemma 4.1.
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 205
/
" " , , ..,. ""
""
."
... . . ".
tI ; ~""., of If'
,
.,., ,,
of".,"
, , • , ,. ,,, t
..
, ,
..
r ,.
"
..
", , ,. ,,, , ,.
..
" "",,,.,,,,, .,,,,,,, ,,,,. ...... i.," , "".,,, , , , ,, ,, , t, ,,.,. "\. .. ... , "''', ,''' " , f' .. ::~ :::::::;:;: :: ~ ~ :: : ~ ~ ~ ~ ~ ~ : ~ : ...
... ;
...
·"
jj
· ,·"."
I I
11 '
·
'
..
,.
'
,.
\\
,
... ", " " ,,,,,., " " 1"""""'4",,,,, ,·,1" """, •• , ,"' ".. 1"""".,,· ."." ••" .,, ". ' """ . . "'1."" ", ", ,."
..
~ • I I ' 'r ••••• ••• I' ~ . t j ".,. ,' r
........
• • • • 14"
.,."""""""
J "'''''i''"ij .,..,.",,,,,,,, " i I. ' iI j
j
j
j
j j
j
t
.,..,."'''''·,, · ' , ; ,
Jjji~~~~~~~~~~~~~~~~~~~~~~Il~~ FIG. 2. Th e directions of the deterministic dynamics F. THEOREM 5.1. If the set N is a determining set of indices, then the Markov semigroup associated to the SOE (2.6) is irreducible. REMARK 5.1. In [1], Agrachev and Sarychev prove something more for the 2D case. Namely, they prove controllability in observed projection, that is, given a spectral threshold N , an initial point uO E L2([0, 27r]2) , with divergence equal to zero, and a time T > 0, there is a control that steers the solution of the infinite dimensional Navier-Stokes equations from uO to a point with assigned finite dimensional projection . As a consequence, approximate controllability holds for the infinite dimensional stochastic POE.
6. A toy model. As an example of how such things go on, we shall examine the following toy model, where we will see how the ideas of th e previous section can work. Consider th e system of stochastic differential equations dx; = - Xt + y; - XtYt + dB t , { dYt = -Yt + x; - XtYt,
(6.1)
where B, is a one dimensional Brownian motion . We will see that the diffusion given by the SDE is hypo-elliptic (so that strong Feller prop erty holds), but the syst em is not controllable. Hence the controllability results on the Navier-Stokes dynamics depend heavily on the geometrical structure of the non-linearity. We can have a rough idea of the deterministic dynamics from Figure 2. It is easy to see that the system has a global in time solution (just apply + Y;). Define the following vector fields Ito formula to
x;
2
a
a
2
F= (-X+Y - xY)ax +(-y+x -xY)ay
and
x=
a ax '
where F is the vector fields given by the deterministic dynamics (the drift) and X is the one given by the direction of the noise. Since
UT,Xl, x]
a
= 2 ay'
206
M. ROMITO
it follows that that the Lie algebra generated by F and X has full rank, once it is evaluated at each point of the state space IR 2 . In other words, Hormander's condition holds. On the other hand, the system is not globally controllable. Indeed, if one consider the solution at the starting point (xo, Yo) = (0,0), it is easy to solve the equation for Yt (it is a linear equation with random coefficients), thus obtaining
which is almost surely non negative. In view of the control theory issues explained in the previous section, one can explain the above phenomenon in the following way: since the direction Ox is the one forced by the noise, both Ox and -Ox are directions that the system is allowed to follow, and Oy = [[F, X], X] as well. On the contrary, the vector field -Oy (which is in the Lie algebra, thus ensuring the regularity of the transition probability densities) is a forbidden direction for the system. REMARK 6.1. The Markov process solution of the system (6.1) is anyway ergodic. Hence, controllability arguments are just sufficient for the proof of uniqueness of the invariant measure. One can use the same argument that is used in E and Mattingly [l l], namely it is sufficient to show that the neighbours of the origin, which is the stable point aitracior of the dynamics, are recurrent for the process. 7. Some numerical results. In this last section we aim to investigate what happens in the case where the hypo-ellipticity condition does not hold. The Navier-Stokes equations, without any external forcing, converge to 0 as the time goes to infinity. If we have an external forcing, namely a white noise forcing, which excites only a few number of modes, it may happen that the other modes are indirectly forced by means of the nonlinearity. This is what essentially happens under the assumptions of the previous sections. Assume now that the forced modes do not constitute a determining set of indices (as defined in Definition 4.1). Intuitively, we believe that in this case the system experiences a sort of decoupling, where the (directly and indirectly) forced modes behave as in the former case, while the others behave as if there were no forcing, hence collapsing to zero. If this picture turns out to be true, it follows that the invariant measure concentrates on a hyper-plane of the state space . 7.1. The numerical simulation. In order to lower the number of equations of the system to be numerically solved, we have chosen to examine the approximation of the 2D Navier-Stokes equations in the simplified
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 207
vorticity" formulation that has been introduced in E and Mattingly [11] . Namely, we consider the following system of SDEs 2
dCk = - vJk l ck +
h.L · I ""' h.L · I k W(ShSl - ChCi) + L W (ChCl + shsd + O"k d{3~' , h+l=k h-l=k ""'
L
2 ""' h.L · I h.L · I k dsk=-vlkl Sk- L W(rhS1+S hCi)+ ""' L W(ShCi-ChSd+'Ykd.B:' , h+l=k h-cleek
where the vorticity is given by
~(t, x) =
L Ck(t) cos(k· x) + Sk(t) sin(k · x) k
and the sums are extended to all pairs k = (k l , k 2 ) of positive integers, each one being less than the spectral threshold N . In order to solve numerically the system of stochastic equation, we have used the backward Euler method (a standard reference for the approximation of stochastic equations is Kloeden and Platen [20]) . In few words , say that the equation to be approximated is y' = f (y) + a dWt , where y is a vector. Then at each time-step,
where b. Wn are independent centred Gaussian random variables, having variance equal to the size of the time step. The implicit equation in the unknown Yn+! is solved using the Newton's method. The problem is stiff, due mainly to the linear part (there are the large negative coefficients -vlkI 2 ) . In order to obtain relevant results we have taken a small viscosity and a comparable intensity of the noise. Different realizations of the simulation have been run , with different initial conditions. Each initial condition has been chosen with a centred st andard normal distribution. Notice th at, even though we know from the theoretical analysis that the Markov process converges for each init ial condition, the same claim can be incorrect if the size of the initial condition is too large (see for example Higham, Mattingly and Stuart [17]) . All the numerical experiments in the following have been run with the following value of the parameters: t> t> t>
Grashof number'' Gr = 50, spectral threshold N = 8, tim e increment b.t = 0.05.
4The vorticity, in the 2D case , is defined as the scalar field given by € = curl u , wher e u is the velocity field. In the Fourier coord inates, if (Uk)kE71 d are the Fourier coefficients of the velocity field, the coefficients of the vorticity are given by €k = ik X Uk . 5The Grashof number is defined (in the 2D case) as the ratio between the intensity of the force and the square of the viscosity (see for example Foias , Manley, Rosa and Tem am [13]).
208
M. ROMITO
'"
17~
'os
'"
'2 5
~ tl t
t
tP
t 11 11
J I
JlJ5
175
t
10!
FIG. 3. The errors graphic of the (1,1) are forced.
Ck
coefficients. The three modes (1,0) , (0,1) and
The first data obtained have been discarded, they should correspond to the transient regime and they don't give any useful information concerning the invariant measure. The time increment is rather small , because of the stiffness of the problem. REMARK 7.1. For the implementation of the numerical method (especially for the random number generator), we have heavily used some libraries from the GNU Scientific Library [37]. 7.2. Conclusions. According to the results of the numerical simulations, the picture we have figured out , seems to be reasonable. The first case we examine is the one where the three modes (1,0), (0,1) and (1,1) are excited. According to the theoretical result (See Theorem 3.3), the invariant measure is fully supported. The empirical averages of the process hence converge to the expectations with respect to the invariant measure. We see that all the modes are excited (see Figures 3 and 4), even though with different intensity (Notice that the size of the standard deviations has been rescaled by a factor .3 for the sake of readability). In the second case, only the mode (1,0) is excited. Most of the energy then stays in the forced mode and in a few others (see Figures 5 and 6). Notice that in this case the dissipation scale is larger than in the previous case, so that the system dissipates the energy faster than in the previous case. Figure 7 summarises and compare the two cases. It represents the intensity of each J c~ + component in the two cases examined, depending on the position in the Fourier space . The largest values correspond to the modes being excited by the noise.
sa
A GEOMETRIC CASCADE FOR THE SP ECT RAL APPROXIMATIO N... 209
t It. tt 1
F IG . 4. Th e errors gmphic of the Sk coefficients. Th e three modes (1, 0), (0, 1) an d (1, 1) are f orced.
FIG. 5 . Th e error s gm phic of the
Ck
coeffi cien ts. Only the mode (1, 0) is f orced.
M. ROMITO
210
FIG. 6. The errors graphic of the
Sk
coeffi cients. Only th e mode (1,0) is fo rced.
FIG. 7. The values of the means for Jc~ + s~. On th e left , only the mode (1, 0 ) is forced. On the right, the three mod es (1,0), (0,1) and (1, 1) are forced.
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION... 211
7.2. It has to be remarked that the numerical simulations we have run , are of little interest from the point of view of turbulence , and their only aim is to give a support to our conjecture. Indeed, the spectral threshold is too small and the main parameter, the Grashof number, corresponds to a not so turbulent regime. REMARK
Acknowledgements. The author wishes to thank D. Blomker, P. Constantin, F . Flandoli, M. Hairer, J. Mattingly, A. Sarychev for the helpful conversations on the subject of this paper . The author wishes also to thank the Institute for Matbematics and its Applications, at the University of Minnesota, for its warm hospitality during the Summer Program on Probability and Partial Differential Equations in Modern Applied Matbematics. REFERENCES [1] A.A. AGRACHEV AND A.V . SARYCHEV, Navier-Stokes equation controlled by degenerate forcing: controllability in finite-dimensional projections, to appear on J . Math. Fluid Mech. [2] J.M . BISMUT, Martingales , the Malliavin calculus and hypo-ellipticity under general Hormander 's conditions, Z. Wahrsch. Verw. Gebiete 56 (1981) , 469-505 [3] S . CERRAI, Second Order PDE's in Finite and Infinite Dimension, Lecture Notes in Math. 1762, Springer Verlag, 2001. [4] P . CONSTANTIN , C . FOlAS, AND R . TEMAM , On the large time Galerkin approximation of the Navier-Stoke s equations, SIAM J. Numer. Anal. 21 (1984) , no. 4, 615--634. [5J P . CONSTANTlN, Geometric statistics in turbulence, SIAM Re view 36(1) (1994) , 73-98. [6] G . DA PRATO AND A. DEBUSSCHE, Ergodicity for the 3D stochastic Navier-Stokes equations, preprint (2003) . [7] G . DA PRATO AND J . ZABCZYK, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Not e Series , 229. Cambridge Unive rsity Press, Cambridge, 1996. [8] J .L. DOOB, Asymptotic properties of Markov transition probabilities, Trans. Amer. Math. Soc. 63 (1948) , 393-421. [9] J .P . ECKMANN AND M. HAIRER, Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, Comm. Math. Phys. 219 (2001) , no . 3, 523-565. [10] D.K. ELWORTHY AND X.M . LI, Formulae for the derivatives of heat semigroups, J . Func. Anal. 125 (1994) , 252-286. [11] W . E AND J .C . MATTlNGLY, Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation , Comm. Pure Appl. Math. 54 (2001), no . 11, 1386-1402. [12] F . FLANDOLI, Irreducibility of the 3-D stochastic Navier-Stokes equation, J. Funct. Anal. 149 (1997), no. 1, 160-177. [13] C . FOlAS, O . MANLEY, R . Rosx , AND R. TEMAM, Navier-Stokes equation s and turbulence, Cambridge University Press, Cambridge, 2001. [14] G. GALLAVOTTI, Foundations of fluid dynamics, translated from the Italian. Texts and Monographs in Physics. Springer Verlag, Berlin, 2002. [15] M . HAIRER, Exponential mixing for a stochastic partial differential equation driven by degenerate noise, Nonlinearity 15 (2002), no. 2, 271-279. [161 R .Z. HAS'MINSKif, Ergodic properties of recurrent diffusion processes and stabilization of the solutions of the Cauchy problem for parabolic equations, Theory Probab. Appl. 5 (1960) ,179-196.
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[17] D.J . HIGHAM, J .C. MATTINGLY, AND A. M. STUART, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101 (2002), no. 2, 185-232. [18] V. JURDJEVIC, Geometric control theory, Cambridge Studies in Advanced Mathematics 51, Cambridge University Press, Cambridge, 1997. [19] V. JURDJEVIC AND I. KUPKA, Polynomial control systems Math. Ann . 272 (1985), no. 3, 361-368. [20] P .E. KLOEDEN AND E . PLATEN, Numerical solution of stochastic differential equations , Applications of Mathematics 23, Springer Verlag, Berlin, 1992. [21] A.N . KOLMOGOROV, Local structure of turbulence in an incompressible fluid at a very high Reynolds number, Dokl. Akad. Nauk SSSR 30 (1941), 299-302 ; English transls., C.R. (Dokl.) Acad . ScL URSS 30 (1941), 301-305, and Proc. Ray. Soc. London Ser . A 434 (1991), 9-13 . [22] L.D . LANDAU AND E .M . LIFSHITZ, Fluid mechanics, Course of Theoretical Physics, VoI. 6 . Pergamon Press, London , Paris, Frankfurt, 1959. [23] J. LERAY, Etude de diverses equations integrales non lineaires et de quelques probiemes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82 . [24] J . LERAY, Essei sur les mouvements plans d'un liquide visqueux que limitent des parois, J . Math. Pures Appl. 13 (1934), 331-418. [25] J . LERAY, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248 . [26] P . MALLlAVIN Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations (Res . Inst . Math. ScL, Kyoto Univ., Kyoto (1976), 195-263 , WHey, New York Chichester Brisbane, 1978. [27J J .C. MATTINGLY, E. PARDOUX, Malliavin calculus and the randomly forced NavierStokes equat ions, preprint (2003). [28] J .C . MATTINGLY, On recent progress for the stochastic Navier-Stokes equations, to appear on Journees Equations aux derivees partielles. [29] S.P . MEYN AND R .L. TWEEDlE, Markov chains and stoch astic stability, Communications and Control Engineering Series. Springer-Verlag London, Ltd ., London, 1993. [30] S.P . MEYN AND R .L. TWEEDlE, Stability of Markovian processes. Il . Continuoustime processes and sampled chains, Adv . in AppI. Probab. 25 (1993), no. 3, 487-517. [31] S.P . MEYN AND R .L. TWEEDlE, Stability of Markovian processes. IlI. FosterLyapunov criteria for continuous-time processes, Adv . in AppI. Probab. 25 (1993), no. 3, 518-548. [32J J . NORRlS, Simplified Malliavin calculus, Seminaire de Probabilites, XX, 1984/85, 101-130, Lecture Notes in Math., 1204, Springer, Berlin, 1986. [33] M. ROMITO, Ergodicity of the finite dimensional approximation of the 3D NavierStokes equations forced by a degenerate noise, J . Stat. Phys. 114 (2004), Nos. 1/2,155-177. [34] D.W . STROOCK, The Malliavin calculus, a function al analytic approach, J. Funct. Anal. 44 (1981), no. 2, 212-257 . [35] D.W. STROOCK, Some applications of stochastic calculus to partial differential equations, Eleventh Saint Flour probability summer school-1981 (Saint Flour, 1981), 267-382, Lecture Notes in Math. 976, Springer, Berlin, 1983. [36] D.W . STROOCK AND S.R.S . VARADHAN, On the support of diffusion processes with applications to the strong maximum principle , Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ . California, Berkeley, Calif., 1970/1971), Vol. Ill: Probability theory, pp. 333-359. Univ. California Press, Berkeley, Calif. , 1972. [37] VV . AA ., The GNU Scientific Library, http;//VWIo7.gnu.org/software/gsl/ .
INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS BJORN SCHMALFUSS· Abstract . The intention of the article is t o show the existence of inert ial manifolds for random dyn amical systems generated by infinite dimensional random evolution equations. To find these manifolds we formulate a random graph transform . This transform allows us to introduce a random dynamical system on graphs. A random fixed point of this system defines the graph of the inertial manifold. In contrast to other publications dealing with these objects we also suppose th at th e linear part of such an evolut ion equation contains random operators. To deal with these objects we apply th e multiplicative ergodic theorem. The key assumption for the existence of an inertial manifold is an w-wise gap condition. Key words. Inertial manifolds, random dyn amical systems, stochastic partial de 's, mult iplicative ergodic theorem. AMS(MOS) subject classifications.
Primary: 37HlO ; secondary: 37H15 ,
60H15, 35B42 .
1. Introduction. Inertial manifolds are objects that allow to interpret the dimension of the long-time dynamics of ordinary or partial differential equations. In the case of a partial differential equation this dyn amics will be finite dimensional. Inertial manifolds are Lipschitz continuous manifolds in the phas e space. On the graph of these manifolds the dynamics is given by an ordinary differential equation of smaller dimension than th e dimension of the original equation. States outside these manifolds will be attracted by th e inertial manifolds exponentially fast . A st andard method to show th e existence of th ese manifolds is the Lyapunov-Perron method. Using this technique the graph of the inertial manifold is given as a fixed point of an operator equation related to our original differential equation, see Chow and Lu [7] , Chueshov [8], Constantin et al.[13], Foias et al. [17] and Temam [23] . Similar methods has been used to find inertial manifolds for sto chasti c (partial) differenti al equations, see Bensoussan and Flandoli [3]' Da Prato and Debussche [14]' Chueshov and Girya [10]' [11]' Duan et al. [16] and for delay equations see Chueshov and Scheutzow [12] and references therein. Another technique for non-autonomous dyn amical syste ms has been introduced by Koksch and Siegmund [18] . In contrast to these applications we will use another method. This method is called graph transform method. This transform allows us to introduce a random dyn amical syst em defined on graphs. A random fixed point for this system defines t he random graph of the random inertial manifold. • Mathematical Institute, University of Paderborn, Warburger StraBe 100, 33098 Pad erborn, Germany (schmalfuss0upb . de) . 213
214
BJORN SCHMALFUSS
This method has been introduced by SchmalfuB [211 and by Duan et al. [15] but only for invariant manifolds. In contrast to all these applications we consider more general random evolution equations. Especially, we are able to treat equations with a random linear part and with non-linearities having a random Lipschitz constant. The curial assumption to find invariant manifolds is the gap condition. We will formulate an w-wisegap condition containing the random Lipschitz constant and random coefficients of the exponential dichotomy condition of the linear part. As a main tool to achieve our results we have to apply the multiplicative ergodic theorem. Since this article is base on techniques from the theory or random dynamical systems we introduce in the next section basic terms from this theory. In Section 3 we introduce the random evolution equation for which we will study random inertial manifolds. Section 4 contains the definition of the random dynamical system on graphs . The fixed point argument giving us the random inertial manifold is described in Section 5. The last section contains examples. 2. Random dynamical systems. In the following we are going to describe the dynamics of systems under the influence of random perturbations. Such a perturbation is given by a noise. The mathematical model for a noise is a metric dynamical system. DEFINITION 2.1. Let (0, .1', JP» be a probability space. Suppose that the measurable mapping B : (IR x 0, B(IR) 0 F)
-+
(0, F)
forms a flow :
Bo = ido , for t, r E IR. The measure JP> is supposed to be invariant (ergodic) with respect to (J. Then the quadro-tuple (0, .1', JP>, (J) is called a metric dynamical system. Metric dynamical systems are generated by the Brownian motion (see Arnold [2] Appendix A) or by the fractional Brownian motion (see Maslowski et al. [19]) or by random impulses (see Schmalfuf [22]). For instance we can choose for the set Co(lR, U) of continuous functions on IR which are zero at zero with values in a separable Hilbert space U. For .1' we choose the associated Borel-a-algebra with respect to the compact open topology, for JP> we choose the Wiener measure Le. the distribution of a Wiener process for some covariance Q and B is given by the Wiener shift :
°
Btw( ·) = w(· + t) - w(t)
for w E 0.
Then we obtain the metric dynamical system (0, .1', JP>, B) which is called the Brownian motion. Indeed JP> is ergodic with respect to B.
INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 215
In the following we will always suppose that we have a metric dynamical system such that JPl is ergodic with respect to B. Let H be some topological space. A random dynamical system with phase space H and with respect to a metric dynamical system is a measurable mapping
cjJ : (]R+ x D x H,B(]R+) ®F ®B(H)) ---. (H,B(H)) satisfying the cocycle property
cjJ(t + T, W, x) = cjJ(t, (}rw, cjJ( T, W, x)),
cjJ(O ,W, x) = x.
The eocycle property is a generalization of the semi-group property. It reflexes the fact that the deterministic dynamics is perturbed by a noise. For our applications we will always suppose that H is a separable Hilbert space with norm 11 . 11 . Later we will allow a slight modification of the measurability of such a system. Generators for random dynamical systems are solution operators for random or stochastic differential equations. For instance , consider the random differential equation with phase space: H = ]Rd
dcjJ dt = f((}tw, cjJ), Suppose that this equation has a globally unique solution for any noise path wand for any initial condition x E ]Rd. This solution at time t is denoted by cjJ(t, w, x). These operators form a cocyclejrandom dynamical system . The measurability of this mapping then follows by some regularity assumptions about the right hand side f . Another tool we need is the multiplicative ergodic theorem. We consider the linear differential equation (2.1) where B is a random matrix. Then this equation generates a linear random dynamical system with time set ]R, see Arnold [2] 3.4.15. We can describe the dynamics of this random dynamical system by the following theorem which is the multiplicative ergodic theorem: THEOREM 2.2. Let B E ]Rd ®]Rd be a random matrix contained in L 1 (D, F, JPl, (}) where (} is supposed to be ergodic. Then there exists a {(}t hEIR -invariant set of full measure, non random numbers pEN, P ::; d, -00 ::; Ap < Ap-l < ... < Al and d 1 , ' " , dp E N and random linear spaces E 1(w),·· · ,Ep(w) of deterministic dimension d1 , ' " ,dp such that
n
(2.2)
]Rd =
E 1(w) EB,, · EB Ep(w)
for wEn. The spaces E, are invariant with respect to the random dynamical system 'IjJ generated by (2.1): 'IjJ(t,w,Ei(w)) = Ei((}tW)
fori = 1,'" ,p,
wEn,
t E]R
216
BJORN SCHMALFUSS
and
)\{} 11'm logll7/J(t,w)xll __ /I,.. if an d only if x E E( i W 0
t_±oo
t
for wEn . The spaces E; depend mea surably on w. In particular, there exist measurable projections on these spaces.
The numbers Ap , ' " ,AI are called Lyapunov exponents to (2.1). The spaces E, are the Oseledets spaces to (2.1). The assertions of the multiplicative ergodic theorem are only true with respect to a {B t hEIR-invariant set of full measure. We can restrict our original metric dynamical system to this set such that B is B(IR) ® F n,F n measurable where the F n trace (J algebra with respect to We will denote this new metric dynamical system by the old symbols (0, F , IP', B). If we can modify our original metric dynamical system in this sense such that some property holds for the new metric dynamical system we say that a property holds B-almost surely, B-a.s. The intention of this article is to find sufficient conditions for the existence of random inertial manifolds for random dynamical systems . We start with basic notations for this purpose . We call a family of random parametrized sets M {M(W)}wEn, M(w) cH a random set if the mapping
n
n.
0:3 w -+ dist(y, M(w)):=
inf
xEM(w)
Ilx -
yll
is a random variable for any y EH. Such a set is called a positively invariant random set with respect to a random dynamical system cP if
cP(t,w ,M(w)) c M(Btw) for t
~
O,W E O.
Let HI, H 2 be a splitting of the Banach space H :
In addition we suppose that there exist continuous projections HI, H 2 . Let
1fl , 1f2
onto
such that w -+ ')'(w ,x) is measurable for fixed x E HI. For fixed w the mapping x -+ ')'( w, x) is Lipschitz continuous . Then we call the set
INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS
217
a random Lipschitz manifold. Indeed M is a random set what follows from Castaing and Valadier [6] Lemma 111.14. In addition, we call a random positively invariant set M exponentially attracting with respect to the random dynamical system 4> if lim
inf
t-oo zEM(8 t w)
114>(t,w,x) - z] = 0
with exponential speed for any x E H. DEFINITION 2.3. An exponentially attracting random Lipschitz manifold which is positively invariant for a random dynamical system 4> is called a random inertial manifold for 4>. The dimension of this manifold is defined by the dimension of HI . Let X be a mapping on 0 with values in jR+. Such a mapping is called tempered if the mapping t --t X (Btw) is subexponentially growing for t --t ±oo B-a.s.: lim log+ X(Btw) = O. t
t-±oo
Note that if X is a random variable on the ergodic metric dynamical system (0, F , P, B) then there exists only one alternative for the above property. Suppose that the random variable X is not tempered. Then we have
(2.3)
. log+ X(Btw) 1im sup =
B- a.s.
00
t
t-±oo
The following lemma is not hard to prove . LEMMA 2.1. (i) Sufficient for temperedness of the random variable X is that lE sup log" X(Btw) <
00.
tE[O,I]
(ii) If X , Y are tempered so is X + Y , X· Y . If 0 ::; X(w) ::; Y(w) and Y is tempered so is X. (iii) X is tempered if and only if Y is tempered where Y(w) = sup X(Bsw). s E [O,I ]
For the proof of (i) see Arnold [2] Page 165. Our intention is to construct a graph of a random inertial manifold as a random fixed point of a particular random dynamical system. DEFINITION 2.4 . A random variable with phase space H is called random fixed point for the random dynamical system 4> if 4>(t,w,X(w)) for t 2: 0 B-a.s.
= X(Btw)
218
BJORN SCHMALFUSS
3. random evolution equations. The objective of our interest will be the dynamics of the following random evolution equation du dt + A(Btw)u = F(lhw, u).
(3.1)
Our task is to formulate conditions which ensure the existence of an inertial manifold . We start with the random linear differential equation
du dt
(3.2)
+ A(Btw)u =
O.
To treat the non-linear evolution equation by the variation of constant formula we have to assume that t ---? -A(Btw) generates a fundamental solution. For the definition see Amann [11 Chapter 11.2. More precisely, we assume that A generates a random dynamical system of linear continuous mappings on H:
U(t + r,w) = U(t,Brw)U(r,w). Details about the existence of such a random dynamical system can be found in Caraballo et al. [41 . For the following we need the exponential dichotomy condition: There exist continuous projections 7rl, 7r2 related to the splitting of the phase space H = HI Ef7 H 2 commuting with U:
We suppose that U (t, W)7rl defined on the finite dimensional space HI is invertible where these inverse mappings are denoted by a negative first argument:
In addition, we assume that there exist random variables aI, a2 E i, (n, F , JID) such that (3.3)
IIU(t,w)7rl ll
::; eJ~ al(O.w)ds
for t ::; 0
IIU(t, W) 7r2 11
::;
eJ~a2(O.w)ds
for t ~ O.
(It follows from assumptions given below that al(w) > a2(w).) We now consider the complete non-linear equation (3.1). Let us assume that W ---?
and that
F(w, x)
is measurable for any x
INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 219
is Lipschitz continuous
(3.4)
117r IF (w, x) - 1r IF(w , y)11
:::; L(w)llx - YII,
111r2F(W, x) - 1r2F(W, y)11
:::; L(w) llx - yll
for any wEn where the Lipschitz constant L(w) depends on w. In particular we assume that this Lipschitz constant is locally integrable:
l
b
L(Bsw)ds < 00 for -
< a < b < 00 .
00
The Lipschitz continuity and the existence of a fundamental solution of the linear problem ensure that the equation
<jJ(t) = U(t,w)x +
I
t
U(t - s, Bsw)F(Bsw , <jJ)ds,
<jJ(O)
=
x EH
has a unique solution for every initial condition <jJ(O) = x . This solution is the mild solution for (3.1). THEOREM 3.1. Suppose that -A is the generator of the random fun-
damental solution U. In addition, we suppose that F satisfies the above measurability and continuity assumption. Then the solution of (3.1) generates a random dynamical system <jJ with phase space H. For our consideration we also need that
(3.5)
sup 111r I F (w, x )1I :::; h(w) , xEH
where
i- , h
sup
111r2F(W,x) 11:::; h(w)
xEH
are tempered random variables.
4. The random graph transform and inertial manifolds. We now introduce a mapping which transforms Lipschitz continuous mappings from HI to H2 into Lipschitz continuous mappings. That transform is defined under the action of the random dynamical system generated by (3.1). Using this transform we will introduce the graph transform. We are going to construct the random inertial manifold as a random fixed point of this mapping. For the following let "Y be a global Lipschitz continuous and bounded mapping from HI into H 2: "Y E C~,I(HI' H2) where
II"Ylle =
sup lI"Y(x+)1I <
00.
x +EH 1
The Lipschitz modulus of "Y is denoted by
11,IILip =
sup xt #xt EH 1
lI"Y(xt ) - "Y(xt ) 11 IIxt - xt ll
220
BJORN SCHMALFUSS
To describe this graph transform we introduce the followingsystem of equations :
w(t) = U(t-T,OTW)Y+ -iT U(t-s ,Osw)7rIF(Osw,w(s)+v(s»ds (4.1)
+
v(t) = U(t,w)-y(w(O))
I
t
U(t - s, OsW)7r2F(OsW, w(s)
+ v(s))ds
for t E [0, T], '"'( E C2,I(Hl, H2), y+ E HI. Suppose for a while that this system has a unique solution (w(·),v(·». We introduce the following notations: cP(T,w,'"'()(y+) := v(T) and 3(T, OTW, '"'()(y+) := w(O). The mapping cP will serve as the random graph transform. At first we have to answer the question when (4.1) has a unique solution. LEMMA 4.1. For every wEn , y+ E H+ and'"'( E C2,I(H1 , H 2) there exists a T(w, II'"YIILip) such that {4.1} has a unique solution (w, v) E C([O,T], HI Ell H2) for 0 T :::; T(w, II'"YIILip). In particular, y+ --+ v(T,w, '"'(, y+) is bounded and continuous. For fixed C > 0 the mapping w --+ T(w , C) is a positive random variable. The mapping
s
T
--+ v(T,
w,'"'(, y+)
is continuous. Proof We introduce the operator C([O, T] ; HI fJ7 H 2 ) 3 (w,v) --+ 7:y,T,w,y+ (w,v) = :
(w, v)
defined by
(4.2)
U(t - T , OTW)Y+ ( U(t,w)-y(w(O))
It U(t-s , Osw)7rIF(Osw, W+V)dS) , t E [0, T].
+ I~ U(t-s ,Osw)7r2F(Osw,w+v)ds
(To evaluate these expressions we have to calculate at first the first equation and then the second one.) According to the Lipschitz continuity of F we obtain for a(w) := lal(w)1 + la2(w)1 sup (lIwI(t) - w2(t)1I tE[O,T]
+ IlvI(t) -
1 T
:::; (eJoT a(08 w )dS
+ eg
(4.3)
a(08
w
)
v2(t)ll)
II7rIilL(Osw)ds(l
1
dS
+ II'"YllLipeg
w a(08 )ds)
T
117r2 1IL(Osw)ds)
x sup (1IwI(t) - w2(t)11
+ IlvI(t) - v2(t)lI)
tE[O,T)
=:
k(T,w, II'"YIILip) sup (1IwI(t) - w2(t)11 tE[O,T]
+ IlvI(t) - v2(t)ll)
INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 221
If T = T(w) is chosen sufficiently small then we see that there exists a contraction constant less than one which is independent of v". The contraction constant k can also be chosen independently of"( if IbllLip is uniformly bounded such that we can write for this constant k(T, w, IbIILip). The mapping Ty,T,w,y+ maps the complete space C([O ,T]; HI EEl H 2 ) into itself. Hence there exists a unique fixed point (w, v). By the independence of the contraction constant of y+ we can see by the parameter version of the Banach fixed point theorem that y+ --t v(T, w, "( )(y+) is continuous . Similarly, we can find the continuity of v* with respect to T. We define T(w, C) by (4.4)
Since T
k(T(w,C),w,C) = --t
(4.5)
1
2'
k(T, w, C) is strongly increasing it follows {w En: T(w , C) ::; t} = {w En : k(t,w,C)
~
1
2} E:F
such that w --t T(w, C) is a random variable. 0 REMARK 4.1. The mapping C --t T(w, C) is decreasing. The following theorem is crucial to find random invariant manifolds. THEOREM 4.2. Let "(*(w ,·) E C~,I(HI' H2 ) where w --t "(*(w, y+) is measurable and satisfies the property
for wEn, y+ E H+. Then "(* is the graph of a random invariant Lipschitz manifold. Proof Let M be the invariant manifold such that for every x+ E H+ we have that x+ + ,,(*(w,x+) E M(w) . The construction of the graph transform implies
+ ,,(*(w,x+)) = 7f1