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On the martingale probem associated to the 2D and 3D Stochastic Navier-Stokes equations

We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension 2, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension 3, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular.

In this article, our aim is to try to improve the understanding of the martingale problems associated to these equations. Let us first set some notations. Let where n is the outward normal to ∂O, and V = (H 1 0 (O)) d ∩H. The norm and inner product in H will be denoted by | · | and (·, ·) respectively. Moreover W is a cylindrical Wiener process on H and the covariance of the noise Q is trace class and non degenerate (see (1.3) and (1.4) below for more precise assumptions).
We also denote by A the Stokes operator in H: where P is the orthogonal projection of (L 2 (O)) 3 onto H and by b the operator b(x, y) = −P ((x · ∇)y), b(x) = b(x, x), x, y ∈ V.
With these notations we rewrite the equations as    dX = (AX + b(X))dt + √ Q dW, (1. 2) We assume that Tr (−A) 1+g Q < ∞, for some g > 0 (1.3) and |Q −1/2 x| ≤ c|(−A) r x|, for some r ∈ (1, 3/2). (1.4) In dimension d = 3, it is well known that there exists a solution to the martingale problem but weak or strong uniqueness is an open problem (see [9] for a survey). However, it has been proved in [4], [7] (see also [11]) that the above assumptions allow to construct a transition semigroup (P t ) t≥0 associated to a Markov family of solutions where the Kolmogorov operator L is defined by for sufficiently smooth functions ϕ on D(A).
In all the article, we choose one Markov family ((X(t, x) t≥0 , Ω x , F x , P x ) as the one constructed in [7].
The fundamental idea in [4] is to introduce a modified semigroup (S t ) t≥0 defined by It can be seen that for K large enough, this semigroup has very nice smoothing properties and various estimates can be proved. Note that, thanks to Feynman-Kac formula, this semigroup is formally associated to the following equation where N is defined for sufficiently smooth functions ϕ on D(A).
In [4], [7], this semigroup is defined only on the Galerkin approximations of (1.2). Let P m denote the projector associated to the first m eigenvalues of A. We consider the following equation This defines, with obvious notations, (P m t ) t≥0 and (S m t ) t≥0 . The following formula holds by a standard argument: Various estimates are proved on (S m t ) t≥0 and transferred to (P m t ) t≥0 thanks to this identity. A compactness argument allows to construct (P t ) t≥0 . Moreover, a subsequence m k can be constructed such that for any x ∈ D(A), (X m k (·, x)) t≥0 converges in law to (X(·, x)) t≥0 .
Note also that similar arguments as in [4] may be used to prove that for smooth ϕ, (S t ϕ) t≥0 is a strict solution to (1.7).
In dimension 2 this result also holds with exactly the same proofs since all arguments for d = 3 are still valid. Note that it is well known that for d = 2 conditions (1.3)-(1.4) imply that, for x ∈ H, there exists a unique strong solution to (1.2) and the proof of the above facts can be simplified.
In the following, we give some properties of the generator of (P t ) t≥0 and (S t ) t≥0 . For d = 2, we explicit a core, identify the abstract generator with the differential operator L on this core and prove existence and uniqueness for the corresponding martingale problem. (See [21] for a similar result). Again, this follows from strong uniqueness but we think that it is interesting to have a direct proof of this fact. Moreover, it can be very useful to have a better knowledge of the Kolmogorov generator and we think that this work is a contribution in this direction. In dimension 3, we are not able to prove this. We explain the difficulties encountered. We hope that this article will help the reader to get a better insight into the problem of weak uniqueness for the three dimensional Navier-Stokes equations. Nonetheless, we explicit a core for the generator of the transformed semigroup (S t ) t≥0 , identify it with the differential operator N on this core and prove uniqueness for the stopped martingale problem. In other words, we prove weak uniqueness up to the time solutions are smooth. Again, this could be proved directly thanks to local strong uniqueness.

The generators
The space of continuous functions on D(A) is denoted by C b (D(A)). Its norm is denoted by · 0 . For k ∈ N, C k (D(A)) is the space of C k functions on D(A). We need several other function spaces on D(A).
Let us introduce the set E 1 ⊂ C b (D(A)) of C 3 functions on D(A) such that there exists a constant c satistying where γ ∈ (1/2, 1] and Note that we identify the gradient and the differential of a real valued function. Also, the second differential is identified with a function with values in L (H). The third differential is a trilinear operator on D(A) and the norm · above is the norm of such operators.
Slightly improving the arguments in [4], it can be proved 1 that P t maps E i into itself and that there exists a constant c > 0 such that Moreover, for f ∈ E 1 , P t f is a strict solution of (1.5) in the sense that it is satisfied for any x ∈ D(A) and t ≥ 0. Again, the result of [4] has to be slightly improved to get this result. In fact, using an interpolation argument, Proposition 5.9 and the various other estimates in [4], it is easy to deduce that, for any , P t f is still a solution of (1.5) but in the mild sense. We define the Ornstein-Uhlenbeck semigroup associated to the linear equation Then it is shown in [4] that For any λ > 0 we set Then since P t f 0 ≤ f 0 , we have Moreover, since P t is Feller, we have by dominated convergence It can be easily deduced that 1 In fact, only Lemma 5.3 has to be improved. In this Lemma, the term L 1 can in fact be estimated in a single step by using Proposition 3.5 of [7] instead of Proposition 5.1 of [4].
It follows classically (see for instance [20]) that there exists a unique maximal dissipative operatorL on C b (D(A)) with domain D(L) such that We recall the following well known characterization of D(L): f ∈ D(L) if and only if ) has a limit for any x ∈ D(A).

Moreover, we have in this casē
Lf Recall also that Similarly, we may define, for k ≥ 0, E k 3 as the space C 3 functions on D(A) such that there exists a constant c satistying where γ ∈ (1/2, 1]. By the various estimates given in [4], it is easy to check that, provided K is chosen large enough, S t maps E k 3 into itself and there exists a constant c > 0 such that Moreover, for f ∈ E k 3 , S t f is a strict solution of (1.7) in the sense that it is satisfied for any x ∈ D(A) and t ≥ 0.
For any λ > 0 we set and prove that there exists a unique maximal dissipative operatorN on ) has a limit for any x ∈ D(A).
Finally, by (2.4), we see that

Construction of cores and identification of the generators
In this section, we analyse the generators defined in the preceeding section.
We start with the following definition.
is a π-core for K if for any ϕ ∈ D(K), there exists a sequence (ϕ n ) n∈N in D which π-converges 2 to ϕ and such that (Kϕ n ) n∈N π-converges to Kϕ.
Let us set G 1 = (λ −L) −1 E 1 for some λ > 0. Clearly for any ϕ ∈ G 1 we have ϕ ∈ D(L) and by (2.3), ϕ ∈ E 1 . Moreover, since (P t ϕ) t≥0 is a strict solution of the Kolmogorov equation. By (2.1) and the definition of E 1 , for any x ∈ D(A) we have is continuous, we have We deduce thatL and G 2 is also a π-core forL. These results hold both in dimension 2 or 3. The problem is that these cores are abstract and strongly depend on the semigroup (P t ) t≥0 . In dimension 3, this is a real problem since we do not know if the transition semigroup is unique. If we were able to construct a core in terms of the differential operator L, this would certainly imply uniqueness of this transition semigroup.
In dimension 2, we are able to construct such a core. Of course, in this case, uniqueness is well known. However, we think that it is important to have explicit cores. This gives many informations on the transition semigroup (P t ) t≥0 .

Theorem 3.2 Let us set
then, in dimension d = 2, H ⊂ D(L) and it is a π-core forL. Moreover, for any f ∈ H , we haveL f = Lf.
The crucial point is to prove the following result.
Proof. Let f ∈ H . By Itô formula applied to the Galerkin equation (1.8), we have for ǫ > 0 We have denoted by L m the Kolmogorov operator associated to (1.8). Since f ∈ H , we have |Lf m (x)| ≤ c(1 + |Ax| 6 ).

(3.2)
It is easy to prove by dominated convergence that when ǫ → 0. Indeed by Lemma 5.3 below, we have as ǫ → 0. The result follows.
It is now easy to conclude the proof of Theorem 3.2. Indeed, by Proposition 3.3, for f ∈ H we have, since P s Lf 0 ≤ Lf 0 , Moreover, since s → P s Lf (x), is continuous for any x ∈ D(A) It follows that f ∈ D(L) andLf = Lf . Finally G 1 ⊂ H and since G 1 is a π-core we deduce that H is also a π-core.

Remark 3.4
We do not use that P t f is a strict solution of the Kolmogorov equation to prove that H ⊂ D(L) andLf = Lf . But we do not know if there is a direct proof of the fact that H is a π-core. We have used that G 1 ⊂ H and that G 1 is a π-core. The proof of G 1 ⊂ H requires (2.1) which is almost as strong as the construction of a strict solution.
Remark 3.5 For d = 3, using Lemma 3.1 in [4], it is easy to prove a formula similar to (3.2) with |(−A) 1/2 X(s, x)| 6 replaced by |AX(s, x)| 4 in the exponential terms. The problem is that where τ * (x) is the life time of the solution in D(A). Thus we are not able to prove Proposition 3.3 in this case.
We have the following result on the operatorN .
Then H k ⊂ D(N) and it is π-core forN . Moreover, for any f ∈ H k we havē The proof follows the same line as above. Indeed, it is easy to use similar arguments as in [4] and prove that for f ∈ E k 3 , (S t f ) t≥0 is a strict solution to (1.7). Arguing as above, we deduce that (λ −N) −1 E k 3 is a π-core forN . Moreover, applying Itô formula to the Galerkin approximations and letting m → ∞ along the subsequence m k -thanks to Lemma 3.1 of [4] to get uniform integrability -we prove, for f ∈ H k , We rewrite this as and deduce as above that f ∈ D(N) andNf = Nf . Finally, since (λ − N ) −1 E k 3 ⊂ H k , we know that H k is also a π-core.

Uniqueness for the martingale problem
Let us study the following martingale problem.   Proof. By a similar proof as for Proposition 3.3, we know that there exists a solution to the martingale problem Uniqueness follows from a classical argument. Let f ∈ E 1 and, for λ > 0 set ϕ = (λ −L) −1 f. Then ϕ ∈ G 1 ⊂ H and Lϕ(η(s))ds is a martingale. Thus, for any solutionP x of the martingale problem, Lϕ(η(s))ds = ϕ(x).

By inversion of Laplace transform we deducẽ
Thus the law at a fixed time t is uniquely defined. A standard argument allows to prove that this implies uniqueness for the martingale problem.
For d = 3 the proof of uniqueness still works. The problem is that we cannot prove existence of a solution of the martingale problem. More precisely, we cannot prove Proposition 3.3.
We can prove existence and uniqueness in d = 3 for the the martingale problem where H is replaced by G 1 , but since the definition of G 1 depends on the semigroup, this does not give any real information.
We have the following weaker result on a stopped martingale problem. and for any is a martingale with respect to the natural filtration and The stopping time τ * is defined by Proof. Existence of a solution for this martingale problem is classical. A possible proof follows the same line as the proofs of Proposition 3.3 and Theorem 3.6, see also Remark 3.5 (see also [9] for more details). In fact, we may choose the Markov family ((X(t, x)) t≥0 , Ω x , F x , P x ) constructed in [7]. It is easy to see that X(t, x) is continuous up to τ * . We slightly change notation and set X(t, x) = X(t ∧ τ * , x). Uniqueness follows from a similar argument as in Theorem 4.3. For ǫ > 0, we define (S ǫ (t)) t≥0 similarly as (S t ) t≥0 but we replace e −K R t 0 |Aη(s)| 2 ds by e −ǫ R t 0 |Aη(s)| 4 ds in (1.6). Proceeding as above, we then define N ǫ ,N ǫ , H ǫ k , and prove that H ǫ k is a π-core forN ǫ and N ǫ ϕ =N ǫ ϕ for ϕ ∈ H ǫ k . LetP x be a solution to the martingale problem and f ∈ E k 3 . For λ, ǫ > 0, we set ϕ = (λ −N ǫ ) −1 , then ϕ ∈ H ǫ k .
By Itô formula -note that in Definition 4.4 it is required that the measure is supported by C([0, T ]; D(A)) -we prove that is also a martingale. We have used : Thus: We multiply by e −λt and integrate over [0, ∞) and obtain, sinceN ǫ ϕ = N ǫ ϕ, By dominated convergence, we may let ǫ → 0 and obtaiñ x))). The conclusion follows.

Technical results
In all this section, we assume that d = 2. Also, for s ∈ R, we set | · | s = |(−A) s · |.
Proof. We first apply Itô's formula to 1 2 |x| 2 (as usual the computation is formal and it should be justified by Galerkin approximations): We deduce, thanks to a classical martingale inequality, where C depends on T, Q, A. It follows that We now apply Itô's formula to 1 4 |x| 4 , We deduce Since √ Q is a bounded operator, using (5.1) we deduce Proof. We apply Itô's formula to and obtain X(t, x)), AX(t, x)) dt We have Since Tr (QA) < ∞, we know that QA is a bounded operator and  The proof of this Lemma follows the same argument as above. It is left to the reader. Then, by the factorization method (see [5]), for any ǫ < g, and we have We take the scalar product with A 2 Y :  (we choose 3 − s/2 < 2 + g and set ǫ = 1 − s/2). The conclusion follows from Lemma 5.3 and by the boundedness of x → −ǫx 6 + cx 4 .