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Strong solutions to SPDEs with monotone drift in divergence form

We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and weak compactness arguments. The result is essentially sharp, in the sense that no extra hypotheses are needed, bar continuity of the nonlinear function in the drift, with respect to the deterministic theory.


Introduction
Let us consider the nonlinear stochastic partial differential equation on L 2 (D), where D ⊂ R n is a bounded domain with smooth boundary. Here γ is the gradient of a continuously differentiable convex function on R n growing faster than linearly at infinity, the divergence is interpreted in the usual variational sense, W is a cylindrical Wiener process, and B is a map with values in the space of Hilbert-Schmidt operators satisfying suitable Lipschitz continuity hypotheses. Precise assumptions on the data of the problem are given in §2 below. Our main result is the well-posedness of (1.1), in the strong probabilistic sense, without any polynomial growth condition on γ nor any boundedness assumption on the noise (see Theorem 2.2 below). The lack of growth and coercivity assumptions on γ makes it impossible to apply the variational approach by Pardoux and Krylov-Rozovskiȋ (see [7,12]), which is the only known general technique to solve nonlinear stochastic PDEs without linear terms in the drift such as (1.1), with the possible exception of viscosity solutions, a theory of which, however, does not seem to be available for such equations. On the other hand, we recall that, if γ is coercive and has polynomial growth, the results in op. cit. provide a fully satisfactory well-posedness result for (1.1).
The available literature dealing with stochastic equations in divergence form such as (1.1) is very limited and, to the best of our knowledge, entirely focused on the case where γ satisfies the above-mentioned coercivity and growth assumptions: see, e.g., [8] and the bibliography of [9] for results on the p-Laplace equation, which corresponds to the case γ(x) = |x| p−1 x, and [13] on stochastic equations in divergence form with doubly nonlinear drift. The main novelty of this paper is thus to provide a satisfactory wellposedness result in the strong sense for such divergence-form equations under neither coercivity nor growth assumptions on γ. On the other hand, it is worth recalling that well-posedness results are available for other classes of monotone SPDEs with nonlinearities satisfying no coercivity and growth conditions, most notably the stochastic porous media equation: see, e.g., [3]. However, the structure of divergence-form equations such as (1.1) is radically different. Indeed, as is well-known, the porous media operator is quasilinear, while the divergence-type operator in (1.1) is fully nonlinear. Moreover, the monotonicity properties (hence the dynamics associated to the the solutions) are different: the porous media operator is monotone in H −1 , whereas the divergence-form operator is monotone in L 2 .
As is often the case in the treatment of evolution equations of monotone type, the first step consists in the regularization of (1.1), replacing γ with its Yosida approximation (a monotone Lipschitz-continuous function), thus obtaining a family of equations for which well-posedness is known to hold (in our case, we also need to add a "small" elliptic term in the drift as well as to smooth the diffusion coefficient B). In a second step, one proves that the solutions to the regularized equations are compact in suitable topologies, so that, by passage to the limit in the regularization parameters (roughly speaking), a process can be constructed that, in a final step, is shown to actually be the unique solution to (1.1) and to depend continuously on the initial datum. It is well known that the last two steps are the more challenging ones, and our problem is no exception.
The approach we follow combines elements of the variational method and ad hoc arguments, most notably a priori estimates on the solutions to regularized equations, weak compactness techniques, and a generalized version of Itô's formula for the square of the norm under minimal integrability assumptions. A crucial role is played by a mix of pathwise and "averaged" 1 a priori estimates. Even though the approach is reminiscent of that in [11], the problem we consider here is of a completely different nature, and, correspondingly, new ideas are needed. In particular, the absence of a linear term in the drift precludes the possibility of applying a wealth of techniques available for semi-linear problems. For instance, the strong pathwise compactness criteria used in op. cit. are no longer available, so that we have to rely on weak compactness arguments only. This way one can construct a limit process, but its identification as a solution expectedly presents major new issues with respect to the case where stronger compactness is available. Moreover, a rather subtle measurability problem arises from the fact that the divergence is not injective, which is the reason for assuming γ to be a continuous monotone map, and not just a maximal monotone graph on R n × R n . A (less regular) solution to the more general problem when γ satisfies only the latter condition will appear elsewhere. We remark that the results obtained here hold under hypotheses that are as general as those of the deterministic theory, except for the continuity assumption on γ (see, e.g., [2, pp. 207-ff.]). 1 That is, in expectation.
Acknowledgments. The authors were partially supported by a grant of The Royal Society. The first-named author is very grateful to Prof. S. Albeverio for the warm hospitality and the excellent working conditions at the Interdisziplinäres Zentrum für Komplexe Systeme, University of Bonn, where parts of this work were written.

Main result
Given a positive real number T , let (Ω, F , (F t ) t∈[0,T ] , P) be a filtered probability space, fixed throughout, satisfying the so-called "usual conditions". We shall denote a cylindrical Wiener process on a separable Hilbert space H by W .
For any two Hilbert spaces U and V , the space of Hilbert-Schmidt operators from U to V will be denoted by L 2 (U, V ). Let D be a smooth bounded domain of R n , and assume that a map is given such that, for a constant C > 0, for all ω ∈ Ω, t ∈ [0, T ], x, y ∈ L 2 (D). To avoid trivial situations, we also assume that, for an x 0 ∈ L 2 (D), B(ω, t, x 0 ) < C for all ω and t. This implies that B grows at most linearly in x, uniformly over ω and t. Furthermore, the map (ω, t) → B(ω, t, x)h is assumed to be measurable and adapted for all x ∈ L 2 (D) and h ∈ H.
We assume that γ is the subdifferential of a continuously differentiable convex function k : R n → R + such that k(0) = 0, lim |x|→∞ k(x) |x| = +∞ (i.e. k is superlinear at infinity), and Then γ : R n → R n is a continuous maximal monotone map, i.e.
γ(x) − γ(y) · (x − y) ≥ 0 ∀x, y ∈ R n (the centered dot stands for the Euclidean scalar product in R n ), and (the graph of) γ is maximal with respect to the order by inclusion. Moreover, the convex conjugate function k * : R n → R + of k, defined as is itself convex and superlinear at infinity. For these facts of convex analysis, as well as those used in the sequel, we refer to, e.g., [6]. All assumptions on B and γ (hence also on k) are assumed to be in force from now on.
Definition 2.1. Let u 0 be an L 2 -valued F 0 -measurable random variable. A strong solution to equation (1.1) is a process u : Ω × [0, T ] → L 2 (D) satisfying the following properties: (i) u is measurable, adapted and u ∈ L 1 (0, T ; W 1,1 0 (D)) (ii) B(·, u)h is measurable and adapted for all h ∈ H and B(·, u) ∈ L 2 (0, T ; L 2 (H, L 2 (D))) P-a.s.; (iii) γ(∇u) is an L 1 (D) n -valued measurable adapted process with (iv) one has, as an equality in L 2 (D), Since γ(∇u) is only assumed to take values in L 1 (D) n , the second term on the lefthand side of (2.1) does not belong, a priori, to L 2 (D). The identity (2.1) has to be interpreted to hold in the sense of distributions, so that the term containing γ(∇u) takes values in L 2 (D) by difference. In fact, the conditions on B in (i) imply that the stochastic integral in (2.1) is an L 2 (D)-valued local martingale.
Let K be the set of measurable adapted processes φ : Our main result is the following.
We do not know whether well-posedness continues to hold also without the condition that the solution belongs to K . This assumption, in fact, plays a crucial role in the proof of uniqueness.
Abbreviated notation for function spaces will be used from now on: Lebesgue and Sobolev spaces on D will be denoted without explicit mention of D itself; for any p ∈ [1, ∞], L p (Ω) will be denoted by Ä p , L p (0, T ) by L p t , and L p (D) sometimes by L p x . Mixed-norm spaces will be denoted just by juxtaposition, e.g. Ä p L q t L r x to mean L p (Ω; L q (0, T ; L r (D))) and L 1 t,x to mean L 1 ([0, T ] × D).

An Itô formula for the square of the norm
We prove an Itô formula for the square of the L 2 -norm of a class of processes with minimal integrability conditions. This is an essential tool to prove uniqueness of strong solutions and their continuous dependence on the initial datum in Sections 5 and 6 below, and it is interesting in its own right.
are measurable adapted processes such that Proof. Note that div ζ ∈ (W 1,∞ 0 ) ′ , hence, by Sobolev embedding theorems and duality, there exists a positive integer r such that div ζ ∈ H −r . Therefore, denoting the Dirichlet Laplacian on L 2 (D) by ∆, there also exists a positive integer m such that (I − δ∆) −m , δ > 0, maps H −r and (a fortiori) L 2 to H 1 0 ∩W 1,∞ . Using the notation h δ := (I−δ∆) −m h, it is readily seen that for all t ∈ [0, T ] P-a.s. as an identity in L 2 , for which Itô's formula yields for all t ∈ [0, T ] P-almost surely. We are going to pass to the limit as δ → 0 in this identity. The dominated convergence theorem immediately implies that, P-a.s., for all t ∈ [0, T ], and y δ 0 2 → y 0 2 , as δ → 0. Defining the real local martingales we are going to show that as δ → 0. In fact, Davis' inequality for local martingales (see, e.g., [10]) yields , and one has, identifying L 2 (H, R) with H and recalling that (I − δ∆) −m is contractive in L 2 , It follows by the Cauchy-Schwarz inequality that the first term on the right-hand side is dominated by , which converges to zero by properties of Hilbert-Schmidt operators and the dominated convergence theorem. Moreover, and, by the Cauchy-Schwarz inequality, again by dominated convergence it follows that To complete the proof, we are going to show that ∇Y δ · ζ δ → ∇Y · ζ in Ä 1 L 1 t,x , which readily implies that in probability for all t ∈ [0, T ]. Since ∇y δ → ∇y and ζ δ → ζ in measure in Ω×(0, T )×D, in view of Vitali's theorem, it suffices to prove that the sequence (∇y δ · ζ δ ) is uniformly integrable in Ω × (0, T ) × D. One has where the second inequality follows by the hypothesis lim sup |x|→∞ k(−x)/k(x) < ∞. By Jensen's inequality for sub-Markovian operators (see [5,Theorem 3.4]) we also have where the right-hand side is uniformly integrable because it converges in Ä 1 L 1 t,x as δ → 0. This yields that (∇y δ · ζ δ ) is uniformly integrable as well, thus concluding the proof.

Well-posedness for an auxiliary SPDE
Let V 0 be a separable Hilbert space, densely and continuously embedded 2 in H 1 0 , and continuously embedded in W 1,∞ . The Sobolev embedding theorem easily implies that such a space exists indeed.
We are going to prove that the auxiliary equation where G is an L 2 (U, V 0 )-valued process, is well posed.
Then equation (4.1) admits a unique strong solution u such that Moreover, the paths of u are P-a.s. weakly continuous with values in L 2 .
The assumptions of Proposition 4.1 are (tacitly) assumed to hold throughout the section.
and consider the regularized equation Since γ λ is monotone and Lipschitz-continuous, it is not difficult to check that the operator v −→ − div γ λ (∇v) + λ∆v satisfies the conditions of the classical variational approach by Pardoux, Krylov and Rozovskiȋ [7,12] on the Gelfand triple H 1 0 ֒→ L 2 ֒→ H −1 , hence there exists a unique adapted process u λ with values in H 1 0 such that

4.1.
A priori estimates. We are now going to establish several a priori estimates for u λ and related processes, both pathwise and in expectation.
We begin with a simple maximal estimate for stochastic integrals that will be used several times in the sequel.
are measurable and adapted processes such that then, for any ε > 0, Proof. By the ideal property of Hilbert-Schmidt operators (see, e.g., [4, p. V.52]), one has where the right-hand side is finite P-a.s. thanks to the assumptions on F and G. Then The proof is finished invoking the elementary inequality and choosing ε properly.
The estimate in the previous lemma will be used only in the case K = R. The more general proof we have given is not more complicated than in the simpler case actually needed.
where u λ in the stochastic integral on the right-hand side has to be interpreted as taking values in L (L 2 , R) ≃ L 2 . Taking supremum in time and expectation we get where, by Lemma 4.2, for any ε > 0. The proof is completed choosing ε small enough and recalling that γ λ is monotone.
Lemma 4.4. The families (∇u λ ) and (γ λ (∇u λ )) are relatively weakly compact in Ä 1 L 1 Proof. Recall that, for any y, r ∈ R n , ones has k(y) + k * (r) = r · y if and only if r ∈ ∂k(y) = γ(y). Therefore, since we deduce, by the definition of γ λ , that By Lemma 4.3 we infer that there exists a constant N , independent of λ, such that Since k * is superlinear at infinity, the family (γ λ (∇u λ )) is uniformly integrable on Ω × Since k is superlinear at infinity, the criteria by de la Vallée Poussin and Dunford-Pettis imply that the sequence ( the relative weak compactness of (∇u λ ) immediately follows by the same property of (γ λ (∇u λ )) proved above.
We shall need below the following classical absolute continuity result, whose proof can be found, for instance, in [2, p. 25].
As customary, both the duality pairing between V and V ′ as well as the scalar product of H have been denoted by the same symbol.
From now on we shall assume, without loss of generality, that λ ∈ ]0, 1].
Proof. Setting v λ := u λ − G · W , equation (4.2) can be written as By Itô's isometry and Doob's inequality, one has 0 . Moreover, since div γ λ (∇u λ ) and ∆u λ belong to Ä 2 L 2 t H −1 , by the previous identity we also deduce that v ′ λ (ω) ∈ L 2 t H −1 for P-a.a. ω ∈ Ω. In particular, taking into account the hypotheses on u 0 and G, there exists Ω ′ ⊂ Ω, with P(Ω ′ ) = 1, such that Let us consider from now on a fixed but arbitrary ω ∈ Ω ′ . Taking the duality pairing of (4.4) by v λ and integrating (more precisely, applying Lemma 4.5) implies that, for all t ∈ [0, T ], Moreover, Young's inequality yields hence also, taking into account the previous estimate, Let k λ be the Moreau-Yosida regularization of k, i.e.
As is well known, k λ is a proper convex function that converges pointwise to k from below, and ∂k λ = γ λ . Therefore, it follows from Taking the supremum with respect to t yields As already observed above, the first three terms on the right-hand side are clearly finite. Moreover, since V 0 ֒→ W 1,∞ , one has by the continuity of k. Since ω was chosen arbitrarily in Ω ′ , the proof is completed.
Let Ω ′ be defined as in the proof of Lemma 4.6, and fix an arbitrary ω ∈ Ω ′ . By for all t ≤ T . Thanks to Young's inequality, convexity of k * , and k * (0) = 0, one has Recalling that k * (γ λ (x)) ≤ γ λ (x) · x for all x ∈ R n , rearranging terms one gets where all terms on the right-hand side are finite, as already established in the proof of Lemma 4.6. Appealing again to the criteria by de la Vallée Poussin and Dunford-Pettis, we immediately infer that (γ λ (∇u λ (ω, ·))) is relatively weakly compact in L 1 t,x . Denoting by M (a constant depending on ω) the right-hand side of the previous inequality, the above estimates also yield This implies, in complete analogy to the previous case, that (I +λγ) −1 ∇u λ is relatively weakly compact in L 1 t,x . Since ∇u λ = λγ λ (∇u λ ) + (I + λγ) −1 ∇u λ , the relative weak compactness of (∇u λ (ω, ·)) in L 1 t,x follows immediately.

Proof of Proposition 4.1.
Let ω ∈ Ω ′ be arbitrary but fixed, where Ω ′ is a subset of Ω with probability one, chosen as in the proof of Lemma 4.6. The relative weak compactness of (γ λ (∇u λ )) in L 1 t,x , proved in Lemma 4.7, implies that there exists η ∈ L 1 t,x such that γ µ (∇u µ ) → η weakly in L 1 t,x , where µ is a subsequence of λ. This in turn implies that √ λu λ is bounded in L 2 t H 1 0 thanks to Lemma 4.6, hence, recalling that ∆ is an isomorphism of H 1 0 and H −1 , λ∆u λ → 0 in L 2 t H −1 as λ → 0, in particular for all t ∈ [0, T ] as λ → 0. Therefore, considering the regularized equation and passing to the limit as µ → 0, we infer that u µ (t) → u(t) weakly in V ′ 0 for all t ∈ [0, T ], hence one can write Moreover, since, thanks to Lemma 4.6, (u µ (t)) is bounded in L 2 , we also have u µ (t) → u(t) weakly in L 2 . In fact, let ε > 0 and ψ ∈ L 2 be arbitrary. Since V 0 is dense in L 2 , there exists φ ∈ V 0 with ψ − φ < ε, and one can write where the second term on the right-hand side converges to zero as µ, ν → 0, and so that, recalling that Hilbert spaces are weakly sequentially complete, u µ (t) converges weakly in L 2 , necessarily to u(t), for all t ∈ [0, T ]. This also immediately implies that u ∈ L ∞ t L 2 x . From this, together with u ∈ C t V ′ 0 , it follows in turn that u ∈ C w ([0, T ]; L 2 ) by a criterion due to Strauss (see [14, Theorem 2.1] -here and below C w ([0, T ]; E) stands for the space of space of weakly continuous functions from [0, T ] to a Banach space E). Furthermore, since all terms in (4.6) except the second one on the left-hand side take values in L 2 , it follows that (4.6) is satisfied also as an identity in L 2 .
Let us show that u ∈ L 1 t W 1,1 0 : the relative weak compactness of (∇u λ ) in L 1 t,x , proved in Lemma 4.7, implies that there exists v ∈ L 1 t,x such that, along a subsequence of λ which can be assumed to coincide with µ, ∇u µ → v weakly in L 1 t,x . Taking into account that u µ ∈ H 1 0 for all µ and that u µ → u weakly* in L ∞ t L 2 x , it easily follows that v = ∇u a.e. in [0, T ] × D and that u ∈ L 1 t W 1,1 0 . As a next step, we are going to show that η = γ(∇u) a.e. in (0, T ) × D. For this we shall need the "energy" identity proved in the following lemma.
x , and f ∈ L 2 t V 0 with f (0) = 0. Furthermore, assume that there exists c > 0 such that Then Proof. The proof if analogous to that of Proposition 3.1, of which we borrow the notation and the setup. In particular, let m ∈ N be such that hence, by Lemma 4.5, x , we have that, up to selecting a subsequence, Therefore, taking Vitali's theorem into account, the lemma is proved if we show that ζ δ · ∇(y δ − f δ ) is uniformly integrable: one has, by Young's inequality and convexity, as well as It follows by Jensen's inequality for sub-Markovian operators, recalling that (I − δ∆) −m and ∇ commute, that where k(c∇y) and k * (cζ) belong to L 1 t,x by assumption, and the same holds for k(c∇f )+ k(c(−∇f )) because f ∈ W 1,∞ . Moreover, the hypothesis lim sup |x|→∞ k(−x)/k(x) < ∞ implies that therefore, taking into account that (I − δ∆) −m is a contraction in L 1 , we obtain that c 2 |ζ δ · ∇(y δ − f δ )| is dominated by a sequence that converges in L 1 t,x , which immediately implies that ζ δ · ∇(y δ − f δ ) is uniformly integrable in [0, T ] × D.
As in the proof of Lemma 4.6, it follows from (4.4) and Lemma 4.5 that This immediately implies by the weak lower semicontinuity of the norm and the weak convergence of u µ (t) to u(t) in L 2 . Moreover, recalling that γ µ (∇u µ ) → η weakly in L 1 t,x and ∇(G · W ) ∈ L ∞ t,x , as The last term on the right-hand side of (4.7) converges to zero as µ → 0 because (∇u µ ) is bounded in L 1 t,x and ∇(G · W ) ∈ L ∞ t,x . We have thus obtained lim sup By Lemma 4.8 we have Moreover, since for all x ∈ R n , we obtain lim sup where (I + µγ) −1 ∇u µ → ∇u and γ µ (∇u µ ) → η weakly in L 1 t,x . In particular, the weak lower semicontinuity of convex integrals yields where N = N (ω) is a constant. Recalling that γ µ ∈ γ((I + µγ) −1 ) and γ = ∂k, we have From this it follows, again by the weak lower semicontinuity of convex integrals, that Let A be an arbitrary Borel subset of (0, T ) × D, z 0 ∈ R n , R > 0 a constant, and where T R : R n → R n , is the truncation operator Then ζ R ∈ L ∞ t,x , and where T R (∇u) → ∇u and k(T R (∇u)) → k(∇u) a.e. in (0, T ) × D as R → ∞, as well as (the latter inequality follows by the assumptions on the behavior of k at infinity). Since k(∇u), k * (η) ∈ L 1 t,x , the dominated convergence theorem implies that for arbitrary z 0 and A, hence also that a.e. in (0, T ) × D for all z 0 ∈ R n . By definition of subdifferential it follows immediately that η = γ(∇u) a.e. in (0, T ) × D.
Let us now show, still keeping ω fixed, that the limit u constructed above is unique. In particular, since η = γ(∇u), it is also unique. Assume that there exist u 1 , u 2 such that To this aim, it suffices to note that, by Lemma 4.8, The monotonicity of γ immediately implies v = 0, i.e. u 1 = u 2 , so that uniqueness of u is proved.
The process u has been constructed for each ω in a set of probability one via limiting procedures along sequences that depend on ω itself. Of course such a construction, in general, does not produce a measurable process. In our situation, however, uniqueness of u allows us to even prove that u is predictable. The following simple observation is crucial: we have proved that from any subsequence of λ one can extract a further subsequence µ, depending on ω, such that u µ converges to u as µ → 0, in several topologies, and that the limit u is unique. This implies, by a classical criterion, that the same convergences hold along the original sequence λ, which does not depend on ω. In particular, u λ (ω, t) → u(ω, t) weakly in L 2 for all t ∈ [0, T ] and for P-a.s. ω. Let us show that u λ → u weakly in Ä 1 L 1 t L 2 x : for an arbitrary φ ∈ Ä ∞ L ∞ t L 2 x , we have u λ (ω, t), φ(ω, t) −→ u(ω, t), φ(ω, t) a.e. in Ω × [0, T ], as well as t,x by Lemma 4.3. Then u λ , φ is uniformly integrable in Ω × [0, T ] by the criterion of de la Vallée Poussin, Mazur's lemma (see, e.g., [4, p. 360]) implies that there exists a sequence (ζ n ) of convex combinations of u λ such that ζ n (ω, t) → u(ω, t) in L 2 in P ⊗ dt-measure, hence a.e. in Ω × [0, T ] along a subsequence. Since (u λ ) is a collection of L 2 -valued predictable processes, the same holds for (ζ n ), so that the P ⊗ dta.e. pointwise limit u of (a subsequence of) ζ n is an L 2 -valued predictable process as well. We also have that u ∈ Ä 2 L ∞ t L 2 x , as it follows by u λ → u in Ä 1 L 1 t L 2 x and the boundedness of (u λ ) in Ä 2 L ∞ t L 2 x .
Moreover, recalling that ∇u λ → ∇u and γ λ (∇u λ ) → η weakly in L 1 t,x P-a.s., and that, by Lemma 4.4, (∇u λ ) and (γ λ (∇u λ )) are bounded in Ä 1 L 1 t,x , an entirely analogous argument shows that ∇u λ → ∇u and γ λ (∇u λ ) → η = γ(∇u) weakly in Ä 1 L 1 t,x . This implies that η is a measurable adapted process, as well as, by weak lower semicontinuity of the norm, We can hence conclude that where, by the weak lower semicontinuity of convex integrals and (I + λγ) −1 ∇u λ → ∇u, P-a.s., hence, by Fatou's lemma, (4.8) Remark 4.9. The proof of uniqueness of u does not depend on γ being single-valued. In particular, all results on u obtained thus far, including the predictability of u, can be obtained under the more general assumption that γ is an everywhere defined maximal monotone graph on R n × R n , with γ = ∂k. However, in this more general framework, the uniqueness of η does not follow, because the divergence is not injective. This implies that we would not be able even to prove that η is a measurable process (with respect to the product σ-algebra of F and the Borel σ-algebra of [0, T ]).

Well-posedness with additive noise
We are now going to prove well-posedness for the equation where G is no longer supposed to take values in L 2 (H, V 0 ), as in the previous section, but just in L 2 (H, L 2 ). In other words, we are considering equation (1.1) with additive noise.
Proposition 5.1. Assume that u 0 ∈ Ä 2 L 2 x is F 0 -measurable and that G : Ω × [0, T ] → L 2 (H, L 2 ) is measurable and adapted. Then equation (4.1) is well posed in K . Moreover, its solution is pathwise weakly continuous with values in L 2 .
Proof. Since one has (I − ε∆) −m : L 2 → H 2m ∩ H 1 0 for any m ∈ N, choosing m > 1/2 + n/4, the Sobolev embedding theorem yields H 2m ֒→ W 1,∞ , hence V 0 := H 2m ∩ H 1 0 satisfies all hypotheses stated at the beginning of the previous section. In particular, setting the ideal property of Hilbert-Schmidt operators implies that G ε is an L 2 (H, V 0 )-valued measurable and adapted process such that It follows by Proposition 4.1 that, for any ε > 0, there exists a unique predictable process In complete analogy to the previous section, the equation in admits a unique (variational) strong solution u ε λ for any ε > 0 and λ > 0. Taking into account the monotonicity of γ λ , Itô's formula yields, for any δ > 0, Taking supremum in time and expectation, it easily follows from Lemma 4.2 that For arbitrary fixed ε, δ > 0, the proof of Proposition 4.1 shows that P-a.s. as λ → 0, and the same holds replacing ε with δ. In particular, on a set of probability one, u ε λ − u δ λ → u ε − u δ weakly* in L ∞ t L 2 x as λ → 0, hence the weak* lower semicontinuity of the norm and Fatou's lemma imply It follows by the ideal property of Hilbert-Schmidt operators, the contractivity of (I − ε∆) −m , and the dominated convergence theorem, that x , hence there exists a predictable L 2 -valued process u such that u ε converges (strongly) to u in Ä 2 L ∞ t L 2 x as ε → 0. Moreover, by (4.8) there exists a constant N , independent of ε, such that where we have used again the ideal property of Hilbert-Schmidt operators and the contractivity of (I − ε∆) −m in the last step. The sequences (∇u ε ) and (γ(∇u ε )) are hence uniformly integrable on Ω × [0, T ] × D by the criterion of de la Vallée Poussin, hence relatively weakly compact in Ä 1 (L 1 t,x ) by the Dunford-Pettis theorem. Therefore, passing to a subsequence of ε, denoted by the same symbol, there exist v and η such that ∇u ε → v and γ(∇u ε ) → η weakly in Ä 1 L 1 t,x as ε → 0. It is then straightforward to check that v = ∇u and 0 . An argument based on Mazur's lemma, entirely analogous to the one used in the proof of Proposition 4.1, shows that η is an L 1 -valued adapted process.
We can now pass to the limit as ε → 0 in (5.2). The strong convergence of u ε to u in in probability as ε → 0. Let φ 0 ∈ V 0 be arbitrary. Since V 0 ֒→ L ∞ , one has u ε (t), φ 0 → u(t), φ 0 in probability for almost all t ∈ [0, T ]. Let us set, for an arbitrary but fixed t ∈ [0, T ], weakly in Ä 1 as ε → 0. Doob's maximal inequality and the convergence In particular, since φ 0 ∈ V 0 and t ∈ [0, T ] are arbitrary, we infer that Recalling that η ∈ L 1 t,x , which implies in turn that div η ∈ L 1 t V ′ 0 , it follows that all terms except the first on the left-hand side have trajectories in C t V ′ 0 , hence that the identity holds for all t ∈ [0, T ]. Moreover, thanks to Strauss' weak continuity criterion, u ∈ C t V ′ 0 and u ∈ L ∞ t L 2 x imply u ∈ C w ([0, T ]; L 2 ). Note also that all terms bar the second one on the left-hand side are L 2 -valued, hence the identity holds also in L 2 for all t ∈ [0, T ].
Let u 01 , u 02 ∈ Ä 2 L 2 x be F 0 -measurable, and G 1 , G 2 : Ω × [0, T ] → L 2 (H, L 2 ) be measurable adapted processes such that from which uniqueness and Lipschitz-continuous dependence on the initial datum follow immediately. We shall actually obtain this estimate as a special case of a more general one that will be useful in the next section: setting where, by monotonicity of γ, ζ α · ∇y α = e −2α· γ(∇u 1 ) − γ(∇u 2 ) · (∇u 1 − ∇u 2 ) ≥ 0. Therefore, taking the supremum in t and expectation on both sides, one has E sup t≤T y α (t) 2 + α E Appendix A. A remark on uniform integrability The classical characterization of uniform integrability by de la Vallée Poussin states that, in the setting of a measure space (X, A) endowed with a finite measure µ, a bounded subset G of L 1 (X, µ; R n ) is uniformly integrable if and only if there exists a continuous increasing convex function ϕ : R + → R + , with ϕ(0) = 0 and lim x→∞ ϕ(x)/x = ∞, such that A ϕ(|g|) dµ < 1 ∀g ∈ G (see, e.g., [1, p. 12]).
The following criterion for uniform integrability can be proved in the same way (the proof is included for completeness). Let G be a subset of L 0 (X, µ; R n ) such that F (G ) is bounded in L 1 (X, µ). Then G is uniformly integrable.
Proof. We have to prove that G is bounded in L 1 (X, µ) and that for any ε > 0 there exists δ > 0 such that, for any A ∈ A with µ(A) < δ, A |g| dµ < ε ∀g ∈ G .
By definition of limit, for any M > 0 there exists R (depending on M ) such that |x| < F (x)/M for all x ∈ R n such that |x| > R. Then for all g ∈ G . Choosing A = X, this proves that G is bounded in L 1 (X, µ). Let ε > 0 be arbitrary, and choose M such that the second-term on the right-hand side of the last inequality is smaller than ε/2. Then δ := ε/(2R) satisfies the required condition.