Asymptotic behavior of stochastic PDEs with random coefficients

We study the long time behavior of the solution of a stochastic PDEs with random coefficients assuming that randomness arises in a different independent scale. We apply the obtained results to 2D- Navier--Stokes equations.


Introduction and setting of the problem
In this work, we consider partial differential equations driven by two different sources of randomness. One is of white noise type and the other one is smoother. This kind of problem is very natural. For instance, if a physical system is submitted to external random forces and if these have different time scales the fast ones can often be approximated by a white noise.
We thus study the following stochastic differential equation,    dX = (AX + b(X) + g(X, Y )dt + σ(X, Y ))dW (t), The two sources of randomness are the Wiener process W and the process Y . We assume that they are independent. More precisely, we are given two separable Hilbert spaces H and K and two filtered probability spaces (Ω, F , P, (F t ) t≥0 ) and ( Ω, F , P, ( F t ) t≥0 ). We shall set E = E P , E = E P . Then W (t) is a cylindrical Wiener process on H associated to the stochastic basis (Ω, F , P, (F t ) t≥0 ) and Y (t) is a K-valued Markov stationary process associated to the stochastic basis ( Ω, F , P, ( F t ) t≥0 ) independent of W .
Several problems can be written as equation (1.1). For instance, it may describe the evolution of a fluid and equation (1.1) is then an abstract form of the Navier-Stokes equation. Other models are reaction-diffusion equations, Ginzburg-Landau equations and so on.
In [3], [4], the case when Y is deterministic and periodic in time has been studied. Since X is not an homogeneous Markov process, the notion of invariant measure does not make sense anymore. Instead, the longtime behaviour is described by an evolutionary system of measures (µ t ) t∈R . It is periodic and is such that if the law of X at time s is µ s then it is µ t at time t. In fact, this evolutionary system can be constructed by disintegration of an invariant measure of an enlarged system which is Markov. Moreover, uniqueness, ergodicity and mixing properties have been generalized to this context.
Our aim in this work is to generalize the results obtained in [3] to more general driving forces. We also define systems of measures which generalize the concept of invariant measures and describe the longtime behaviour. It is also obtained by disintegration of an invariant measure but in a more complicated way.
We assume that (1.1) has a unique continuous solution. This is the case in the examples described above if the function g(X, Y ) is Lipschitz in X, has polynomial growth in Y and Y has moments for instance. We denote the solution by X(t, s, x).
We also assume that there is a continuous Markov process Y (t, s, y), t ≥ s, y ∈ K on ( Ω, F , P, ( F t ) t≥0 ) such that Y (s, s, y) = y and A typical example is when Y is the solution of a stochastic partial differential equation driven by another Wiener process and Y is a stationary solution of this equation. The simplest case is given by an Ornstein Uhlenbeck process: where B : D(B) ⊂ K → K is self-adjoint, strictly negative and such that B −1 is of trace class and V is a cylindrical Wiener process in R with values in K independent of W . We set . Obviously, P s,t ϕ(x) is a random variable in ( Ω, F , P) and for each ω ∈ Ω it holds P s,r P r,t = P s,t , s ≤ r ≤ t.
One should not confuse P s,t ϕ(x) with E[P s,t ϕ(x)]. The latter does not fulfil the cocycle law since X(t, s, x) is not a Markov process in general.
In this paper we are interested in the asymptotic behavior of P s,t ϕ(x). We shall proceed as follows. First we construct an enlarged homogeneous Markov process Z(t, s, x, h) with state space H × K , where K := C((−∞, 0]; K).
Then assuming that Z possesses an invariant measure ν(dx, dh) we show that a disintegration of ν produces a family (µ t ) t∈R of random probability measures on H such that ω a.s.
Such a family of measures is called an evolutionary system of measures. Then, we give sufficient conditions for uniqueness of an evolutionary system of measures. We generalize the classical criterion based on irreducibilty and strong Feller property. We also show that the recent method developed in [10] generalizes to our context. Finally, we illustrate our results on the two dimensional Navier-Stokes.
It is not difficult to check that It follows easily that H is a Markov process which is clearly homogeneous. We assume that for any h ∈ K the following equation has a unique continuous solution Then we define the spaces (we shall denote by E 1 the expectation in this space) and and we consider the homogeneous Markov process Z(t, s, x, h) = (X h (t, s, x), H(t, s, h)).
We denote by Q t , t ≥ 0, and R t , t ≥ 0 the transition semigroups associated to Z and H respectively. We also denote by P h t,s the transition operators associated to X h .

Stationary processes
If where X(t, s, x) is the solution to (1.1). Moreover H(t, s, h)(θ) = Y (t + θ) so that where, for a function f defined on (−∞, t], τ t f is the function defined on (−∞, 0] by τ t f (θ) = f (t + θ). It follows that Note also that the stochastic process H(t, s, h) is stationary since the laws of τ t Y | (−∞,t] and τ s Y | (−∞,s] coincide for any t, s. Moreover, it does not depend on s. We shall denote it by H, and by λ its invariant law.
Conversely, let H be a stationary process associated to H. Then Since Y is clearly stationary, it follows that it is a stationary process associated to Y .
This shows that we have a one to one correspondance between stationary processes for H and Y .
Let λ be an invariant measure for H: If we denote by λ 0 the image measure of λ by the mapping Therefore λ 0 is an invariant measure for Y . We now prove that ergodicity is transferred to H. In the proof of this result, we also prove that the law of the process Y determines the law of H. Proposition 2.1 Assume that Y has a unique invariant measure, then the process H is stationary and ergodic.
Proof. Let λ be an invariant law of H. Assume that where η ∈ [0, 1] and λ 1 , λ 2 are invariant for H. Given θ ∈ R consider the mapping and denote by λ θ , λ 1 θ , λ 2 θ the image measures of λ, λ 1 , λ 2 respectively by this mapping. By the discussion above, λ 0 , λ 1 0 , λ 2 0 are invariant for Y . Therefore We have used the fact that λ 0 is the image law of λ and the invariance of λ 0 .

Evolutionary systems of measures
We now define the main new object of our article. We expect that the long time behaviour of the solutions (1.1) is described by a random family of measures (µ t ) t∈R satisfying P a.s.
for all ϕ ∈ C b (H), t > s. This is a natural generalisation of the notion of invariant measure. This definition is too general and we need to make some restrictions on the system (µ t ) t∈R, ω∈ Ω . Without loss of generality, we assume that the forcing is associated to a shift (τ t ) t∈R : We consider the stationary process (H(t)) t∈R in K constructed above and defined by H(t) = τ t Y | (−∞,t] and denote by λ its invariant measure. Then clearly We assume that (µ t ) t∈R, ω∈ Ω satisfies the following consistency assumption: This says that two solutions of (1.1) with the same past define the same evolution.
It is also natural to assume that, for each t ∈ R, µ t is measurable with respect to the σ-field generated by Y (s), s ≤ t.
A random family of measures (µ t ) t∈R satisfying these properties is called an evolutionary system of measures.

Invariant measures for Z are associated to evolutionary systems of measures
Let us assume that Z possesses an invariant measure ν: Therefore λ is invariant for H. Let H be a stationary process associated to H with invariant law λ. We set Theorem 3.2 The family (µ t ) t∈R is an evolutionary system of measures.
Proof. We prove that (2.7) holds. The other properties of evolutionary systems are clearly satisfied.
Moreover, by the invariance of λ for the process H we get This yields Since H(s) has law λ, H(t, s, H(s)) = H(t) and X H(s) (t, s, x) = X(t, s, x), we can rewrite (3.3) as (their are functions of H(t)) and we deduce by the arbitrariness of α that However, the set of all ω ∈ Ω for which identity (2.7) holds depends of t, s, ϕ. Modifying the disintegration of ν, we can easily get rid of this dependence. We proceed as in [3]. First, taking ϕ = 1l C for C in a countable set generating Borel sets of H, we can show that (2.7) holds in fact for every ϕ ∈ B b (H) for almost every ω depending only on t, s. From Fubini's theorem we find that P-a.s.
for almost all t ≥ s n and set µ n t = P * sn,t µ sn , t ≥ s n . From the continuity of t → X(t, s, x) for almost every ω, we deduce that t → µ n t is continuous. Moreover µ n t = µ n+1 t , a.s. t ≥ s n , so that by continuity By the continuity in t we deduce for all t ∈ R P * sn,t µ sn = µ t , ∀ t ≥ s n and for t ≥ s ≥ s n we have Therefore we can conclude that P-a.s.
as claimed. Since µ t = µ t for almost all t ∈ R, the consistency relation is still satisfied by µ t . Let us now see how an evolutionary system of measures ( µ t ) t∈R, ω∈ Ω yields an invariant measure for Z. By definition, µ t is measurable with respect to the σ-field generated by Y (s), s ≤ t. This implies that there exists a measurable function h → µ t h such that µ t = µ t H(t) . By the consistency assumption (2.8), we have P-a.s. for t ≥ s in a set I of full measure, so that µ t does not depend on t for t ∈ I. We choose s 0 ∈ I, set µ h = µ s 0 h and define ν(dx, dh) = µ h (dx)λ(dh).
Then, since L (Y (·, s 0 )) = λ and X H(s 0 ) (t, s 0 , x) = X(t, s 0 , x), we have for Therefore, for ψ of the form ψ(x, h) = ϕ(x)α(h), Since the left hand side is a continuous function of t, we deduce that the equality holds for all t ∈ R and that ν is an invariant measure for Z. It follows from our discussion that the correspondance ν → ((µ t ) t∈R , λ) is a bijection. In particular, if there exists a unique invariant measure with marginal λ for Z, there exists a unique evolutionary system of measure.

A simple example
We consider the following special form of (1.1) in R.
where Y is the stationary process and V , W are independent Wiener processes. We consider the homogeneous Markov process introduced above Z(t, s, x, h) = (X h (t, s, x), H(t, s, h)), t ≥ s, x ∈ H, h ∈ K .
Let us write explicitly Z(t, s, x, h). We have This implies that the probability measure on H × K , is the unique invariant measure for Z. In other words we have Let ν(dx, dh) = µ x (dx)λ(dh) be a disintegration of ν. Then we have λ = L (H(0)) and We see on this simple example that it is necessary to parametrize the evolutionary system of measure by the whole history of the driving process.

Ergodicity in the regular case
We assume for simplicity in this section that the Markov process Y has a unique invariant measure. It is then the invariant law of Y and Y is ergodic. We construct the ergodic process H as above and denote by λ its invariant law.
We generalize the famous Doob criterion (see for instance [5]) of ergodicity to evolutionary system of measures.
We say that P s,t is regular at ω if for any s < t we have π ω s,t (x, ·) ∼ π ω s,t (y, ·) for all x, y ∈ H.
We notice the following straightforward identity, π s,t (x, E) = H π s,s+h (x, dy)π s+h,t (y, E), s + h < t, h > 0, E ∈ B(H). Proposition 5.1 If the transition semigroup P s,t is strong Feller and irreducible at ω, then it is regular at ω.
Proposition 5.2 Assume that P s,t is regular at ω and that it possesses an invariant set of probabilities µ t , t ∈ R. Then µ t ( ω) is equivalent to π ω s,t (x, ·) for all s < t and x ∈ H.
Next we prove the following theorem which can be used in several applications provided the noise W is non degenerate. Theorem 5. 3 We assume that P s,t is regular for almost all ω and that the Markov process Y has a unique invariant measure. Then Q t has at most one invariant measure which is in addition ergodic. It follows that there exists a unique evolutionary system of measures.
Proof. It suffices to prove that all invariant measures of Q t are ergodic. Let ν(dx, dh) = µ h (dx)λ(dh) be an invariant measure for Q t and Γ ∈ B(H ) be an invariant set for ν:

Then for any
On the other hand, We have proved that, P-a.s., Similarly we show that µ H(t) (Γ H(t) ) = 0 or 1, ∀ t ∈ R.

Remark 5.4
The proof of Theorem 4.3 in [3] is not complete. In fact in that theorem we have proved only that ν h (Γ h ) = 0 or 1. One has to use the same argument as before to arrive at the conclusion.

Uniqueness by asymptotic strong Feller property
We assume again that Y is ergodic and that the process Y (·, s, y) has a unique invariant measure. The following definition is a natural generalization to P s,t of a concept introduced in [10].
Definition 6.1 We say that P s,t is asymptotic strong Feller (ASF) at x ∈ H if there is a sequence of pseudo-metrics {d n } on H such that W dn (π s,s+tn (x, ·), π s,s+tn (y, ·)) = 0. (6.2) P s,t is called asymptotically strong Feller (ASF), if it is asymptotically strong Feller at any x ∈ H and s 0 ∈ R.

Remark 6.2 The previous definition is independent of s because by station-
W dn (π s,s+tn (x, ·), π s,s+tn (y, ·)) , is independent of s.
The following result can be proved as in [10].

Proposition 6.3
Assume that for some s > 0 there exist t n ↑ +∞, δ n → 0 and C(s, |x|) locally bounded with respect to |x| and such that for γ < 1, Then P s,t is ASF.

Then we have
Using the same proof of for Lemma 5.6 in [3], we prove the following result.
Lemma 6.5 Let d ≤ 1 be a pseudo-metric on H and let ν 1 and ν 2 be two invariant measures for Q t with the same marginal λ. Let us denote by (µ 1 t ) t∈R and (µ 2 t ) t∈R the system of random measures constructed in Section 3. Then we have W d (π s,t+s (y, ·) − π s,t+s (z, ·)) .

(6.4)
Theorem 6.6 Assume that (P s,t ) t≥s is ASF and that there is Then there exists at most one invariant measure for (Q t ) t≥0 .
By Lemma 6.5 and stationarity Letting n → ∞ yields 2 But this is impossible because µ 1 h and µ 2 h are almost surely singular.

Application to 2D Navier-Stokes equations
We illustrate the above theory on the two-dimensional Navier-Stokes equations on a bounded domain O ⊂ R 2 with Dirichlet boundary conditions and a stationary forcing term. The unknowns are the velocity X(t, ξ) and the pressure p(t, ξ) defined for t > 0 and ξ ∈ O: with Dirichlet boundary conditions and supplemented with the initial condition For simplicity, we assume that f = 0. Following the usual notations we rewrite the equations as,            dX(t) = (AX(t) + b(X(t)) + g(X(t), Y (t)))dt +σ(X(t), Y (t))dW (t), s ≤ t, X(s) = x.

(7.2)
Here A is the Stokes operator P is the orthogonal projection of (L 2 (O)) 2 on H and b the operator Moreover W is a cylindrical Wiener process on a filtered probability space (Ω, F , (F t ) t≥0 , P) in H. Finally, f : R → H is continuous and 2π-periodic with respect to t. We denote by | · | the norm in H, by · the norm in V and by (·, ·) the scalar product in H.
We assume that there exist K 1 > 0 and h : R → R such that Proof. By Itô's formula we deduce that and, by stationarity of Y and (7.3), We deduce from (7.6), for any M > 0, On the other hand, for any ǫ > 0 there exists a compact set and by the stationarity of Y , By the Krylov-Bogoliubov theorem there exists an invariant measure ν for Z.
By Proposition 7.1 we deduce the existence of the family ( µ t ) t∈R, ω∈ Ω . Concerning uniqueness, we can use the criteria derived in sections 5 and 6. For instance, assume that the noise is addiditive : σ(x, y) = √ C and non degenerate in the following sense Tr C < ∞, C −1/2 (−A) −1/2 ∈ L(H). (7.8) Then using the arguments in [1], [2], [6], [9] it is not difficult to prove that the strong Feller property holds and that the transition semigroup is almost surely irreducible. Then by Theorem 5.3 and section 3, there exists a unique evolutionary system of measure. The nondegeneracy assumptions can be weakened using the ASF property. If we consider periodic boundary condition instead of Dirichlet boundary then the argument in [10] section 4.5 can be adapted to our setting and prove that the ASF property holds if for some c 1 , c 2 ≥ 0 and if the noise acts on a sufficiently large number of modes. Then, uniqueness of evolutionary system of measures holds if one can prove that there exists x 0 such that (6.5) holds.
It is probably also possible to extends the more difficult truly elliptic case treated in [10] section 4.6 but this requires much more work and is beyond the scope of this work.

Uniqueness by coupling
In the same spirit as in [3], we show that coupling arguments extend to our situation. We do not consider the most general case which requires lengthy proofs. Instead, we consider a non degenerate noise so that the argument is not too long. The case of degenerate noise treated for instance in [7], [11], [12], [13] could be treated by mixing the arguments in these papers and the ideas below.
We are here concerned with the equation (7.2) with a non degenerate additive noise σ(x, y) = √ C satisfying (7.8). We need a further assumption on the process Y . We assume that it posseses a Lyapunov structure. More precisely that there exists κ 1 , κ 2 > 0 such that for all t ≥ s: Also, for simplicity, we consider the case when (7.3) holds with h(r) = κ 3 (1 + x), x ≥ 0. A different h would require a different Lyapunov structure.
By Ito's formula and Poincaré inequality: where λ 1 is the first eigenvalue of A. Assumption (7.9) implies: By the Markov property, we obtain for t ≥ r ≥ s Recall that F 1 r = F r × F r . Let T ≥ 0 and set then, for k ≥ 0, we obtain and there exists C(α, T ) such that for α < 1 2 κ 5 E 1 (e ατ ) ≤ C(α, T )e αs (1 + |x| 2 ). (7.11) The following Lemma is also proved as in [3, Lemma 6.2].
Proof. Take the supremum in g 0 ≤ 1 and then the expectation E in (7.12).
The result follows taking first K large and then choosing δ small.
Now we can conclude the proof as in [3] with minor modifications.
We are now ready to construct a coupling by proceeding as in the proof of [3, Proposition 6.5].