Global martingale solutions for stochastic Shigesada-Kawasaki-Teramoto population models

The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods for evolution equations. Instead, the existence proof is based on the entropy structure of the model, a novel regularization of the entropy variable, higher-order moment estimates, and fractional time regularity. The regularization technique is generic and is applied to the population system with self-diffusion in any space dimension and without self-diffusion in two space dimensions.


Introduction
Shigesada, Kawasaki, and Teramoto (SKT) suggested in their seminal paper [37] a deterministic cross-diffusion system for two competing species, which is able to describe the segregation of the populations. A random influence of the environment or the lack of knowledge of certain biological parameters motivate the introduction of noise terms, leading to the stochastic system for n species with the population density u i of the ith species: (1) du i − div A ij (u)∇u j · ν = 0 on ∂O, t > 0, i = 1, . . . , n, and diffusion coefficients (3) A ij (u) = δ ij a i0 + n k=1 a ik u k + a ij u i , i, j = 1, . . . , n, where O ⊂ R d (d ≥ 1) is a bounded domain, ν is the exterior unit normal vector to ∂O, (W 1 , . . . , W n ) is an n-dimensional cylindrical Wiener process, and a ij ≥ 0 for i = 1, . . . , n, j = 0, . . . , n are parameters. The stochastic framework is detailed in Section 2. The deterministic analog of (1)-(3) generalizes the two-species model of [37] to an arbitrary number of species. The deterministic model can be derived rigorously from nonlocal population systems [19,35], stochastic interacting particle systems [8], and finite-state jump Markov models [2,13]. The original system in [37] also contains a deterministic environmental potential and Lotka-Volterra terms, which are neglected here for simplicity.
We call a i0 the diffusion coefficients, a ii the self-diffusion coefficients, and a ij for i = j the cross-diffusion coefficients. We say that system (1)- (3) is with self-diffusion if a i0 ≥ 0, a ii > 0 for all i = 1, . . . , n, and without self-diffusion if a i0 > 0, a ii = 0 for all i = 1, . . . , n.
The aim of this work is to prove the existence of global nonnegative martingale solutions to system (1)-(3) allowing for large cross-diffusion coefficients. The existence of a local pathwise mild solution to (1)-(3) with n = 2 was shown in [30,Theorem 4.3] under the assumption that the diffusion matrix is positive definite. Global martingale solutions to a SKT model with quadratic instead of linear coefficients A ij (u) were found in [18]. Besides detailed balance, this result needs a moderate smallness condition on the cross-diffusion coefficients. We prove the existence of global martingale solutions to the SKT model for general coefficients satisfying detailed balance. This result seems to be new.
There are two major difficulties in the analysis of system (1). The first difficulty is the fact that the diffusion matrix associated to (1) is generally neither symmetric nor positive semidefinite. In particular, standard semigroup theory is not applicable. These issues have been overcome in [9,10] in the deterministic case by revealing a formal gradient-flow or entropy structure. The task is to extend this idea to the stochastic setting.
In the deterministic case, usually an implicit Euler time discretization is used [24]. In the stochastic case, we need an explicit Euler scheme because of the stochastic Itô integral, but this excludes entropy estimates. An alternative is the Galerkin scheme, which reduces the infinite-dimensional stochastic system to a finite-dimensional one; see, e.g., the proof of [32,Theorem 4.2.4]. This is possible only if energy-type (L 2 ) estimates are available, i.e. if u i can be used as a test function. In the present case, however, only entropy estimates are available with the test function log u i , which is not an element of the Galerkin space.
In the following, we describe our strategy to overcome these difficulties. We say that system (1) has an entropy structure if there exists a function h : [0, ∞) n → [0, ∞), called an entropy density, such that the deterministic analog of (1) can be written in terms of the entropy variables (or chemical potentials) w i = ∂h/∂u i as (4) ∂ t u i (w) − div where w = (w 1 , . . . , w n ), u i is interpreted as a function of w, and B(w) = A(u(w))h ′′ (u(w)) −1 with B = (B ij ) is positive semidefinite. For the deterministic analog of (1), it was shown in [11] that the entropy density is given by where the numbers π i > 0 are assumed to satisfy π i a ij = π j a ji for all i, j = 1, . . . , n. This condition is the detailed-balance condition for the Markov chain associated to (a ij ), and (π 1 , . . . , π n ) is the corresponding reversible stationary measure [11]. Using w i = π i log u i in (4) as a test function and summing over i = 1, . . . , n, a formal computation shows that A similar expression holds in the stochastic setting; see (29). It provides L 2 estimates for ∇ √ u i if a i0 > 0 and for ∇u i if a ii > 0. Moreover, having proved the existence of a solution w to an approximate version of (1) leads to the positivity of u i (w) = exp(w i /π i ) (and nonnegativity after passing to the de-regularization limit).
To define the approximate scheme, our idea is to "regularize" the entropy variable w. Indeed, instead of the algebraic mapping w → u(w), we introduce the mapping Q ε (w) = u(w) + εL * Lw, where L : D(L) → H with domain D(L) ⊂ H is a suitable operator and L * its dual; see Section 3 for details. The operator L is chosen in such a way that all elements of D(L) are bounded functions, implying that u(w) is well defined. Introducing the regularization operator R ε : D(L) ′ → D(L) as the inverse of Q ε : D(L) → D(L) ′ , the approximate scheme to (1) is defined, written in compact form, as (7) dv(t) = div B(R ε (v))∇R ε (v) dt + σ u(R ε (v)) dW (t), t > 0.
The existence of a local weak solution v ε to (7) with suitable initial and boundary conditions is proved by applying the abstract result of [32,Theorem 4.2.4]; see Theorem 12. The entropy inequality for w ε := R ε (v ε ) and u ε := u(w ε ), ∇w ε (s) : B(w ε (s))∇w ε (s)dxds ≤ C(u 0 , T ), up to some stopping time τ R > 0 allows us to extend the local solution to a global one (Proposition 15). For the de-regularization limit ε → 0, we need suitable uniform bounds. The entropy inequality provides gradient bounds for u ε i in the case with self-diffusion and for (u ε i ) 1/2 in the case without self-diffusion. Based on these estimates, we use the Gagliardo-Nirenberg inequality to prove uniform bounds for u ε i in L q (0, T ; L q (O)) with q ≥ 2. Such an estimate is crucial to define, for instance, the product u ε i u ε j . Furthermore, we show a uniform estimate for u ε i in the Sobolev-Slobodeckij space W α,p (0, T ; D(L) ′ ) for some α < 1/2 and p > 2 such that αp > 1. These estimates are needed to prove the tightness of the laws of (u ε ) in some sub-Polish space and to conclude strong convergence in L 2 thanks to the Skorokhod-Jakubowski theorem.
For the uniform estimates, we need to distinguish the cases with and without selfdiffusion. In the former case, we obtain an L 2 (0, T ; H 1 (O)) estimate for u ε i , such that the product u ε i ∇u ε j is integrable, and we can pass to the limit in the coefficients A ij (u ε i ). Without self-diffusion, we can only conclude that (u ε i ) is bounded in L 2 (0, T ; W 1,1 (O)), and products like u ε i ∇u ε j may be not integrable. To overcome this issue, we use the fact that a ij u ε j and write (1) in a "very weak" formulation by applying the Laplace operator to the test function. Since the bound in L 2 (0, T ; W 1,1 (O)) implies a bound in L 2 (0, T ; L 2 (O)) bound in two space dimensions, products like u ε i u ε j are integrable. In the deterministic case, we can exploit the L 2 bound for ∇(u ε i u ε j ) 1/2 to find a bound for u ε i u ε j in L 1 (0, T ; L 1 (O)) in any space dimension, but the limit involves an identification that we could not extend to the martingale solution concept.
On an informal level, we may state our main result as follows. We refer to Section 2 for the precise formulation.
Theorem 1 (Informal statement). Let a ij ≥ 0 satisfy the detailed-balance condition, let the stochastic diffusion σ ij be Lipschitz continuous on the space of Hilbert-Schmidt operators, and let a certain interaction condition between the entropy and stochastic diffusion hold (see Assumption (A5) below). Then there exists a global nonnegative martingale solution to (1)-(3) in the case with self-diffusion in any space dimension and in the case without self-diffusion in at most two space dimensions.
We discuss examples for σ ij (u) in Section 7. Here, we only remark that an admissible diffusion term is where 1/2 ≤ α ≤ 1, δ ij is the Kronecker symbol, a k ≥ 0 decays sufficiently fast, (e k ) is a basis of the Hilbert space U with inner product (·, ·) U . We end this section by giving a brief overview of the state of the art for the deterministic SKT model. First existence results for the two-species model were proven under restrictive conditions on the parameters, for instance in one space dimension [26], for the triangular system with a 21 = 0 [33], or for small cross-diffusion parameters, since in the latter situation the diffusion matrix becomes positive definite [17]. Amann [1] proved that a priori estimates in the W 1,p (O) norm with p > d are sufficient to conclude the global existence of solutions to quasilinear parabolic systems, and he applied this result to the triangular SKT system. The first global existence proof without any restriction on the parameters a ij (except nonnegativity) was achieved in [22] in one space dimension. This result was generalized to several space dimensions in [9,10] and to the whole space problem in [21]. SKTtype systems with nonlinear coefficients A ij (u), but still for two species, were analyzed in [15,16]. Global existence results for SKT-type models with an arbitrary number of species and under a detailed-balance condition were first proved in [11] and later generalized in [31].
This paper is organized as follows. We present our notation and the main results in Section 2. The operators needed to define the approximative scheme are introduced in Section 3. In Section 4, the existence of solutions to a general approximative scheme is proved and the corresponding entropy inequality is derived. Theorems 3 and 4 are shown in Sections 5 and 6, respectively. Section 7 is concerned with examples for σ ij (u) satisfying our assumptions. Finally, the proofs of some auxiliary lemmas are presented in Appendix A, and Appendix B states a tightness criterion that (slightly) extends [ The inner product of a Hilbert space H is denoted by (·, ·) H , and ·, · V ′ ,V is the dual product between the Banach space V and its dual V ′ . If F : U → V is a Fréchet differentiable function between Banach spaces U and V , we write DF [v] : U → V for its Fréchet derivative, for any v ∈ U.
Given two quadratic matrices A = (A ij ), B = (B ij ) ∈ R n×n , A : B = n i,j=1 A ij B ij is the Frobenius matrix product, A F = (A : A) 1/2 the Frobenius norm of A, and tr A = n i=1 A ii the trace of A. The constants C > 0 in this paper are generic and their values change from line to line.
Let (Ω, F , P) be a probability space endowed with a complete right-continuous filtration F = (F t ) t≥0 and let H be a Hilbert space. Then L 0 (Ω; H) consists of all measurable functions from Ω to H, and L 2 (Ω; H) consists of all H-valued random variables v such that Let U be a separable Hilbert space and (e k ) k∈N be an orthonormal basis of U. The space of Hilbert-Schmidt operators from U to L 2 (O) is defined by and it is endowed with the norm F L 2 (U ;L 2 (O)) = ( ∞ k=1 F e k 2 L 2 (O) ) 1/2 . Let W = (W 1 , . . . , W n ) be an n-dimensional U-cylindrical Wiener process, taking values in the separable Hilbert space U 0 ⊃ U and adapted to the filtration F. We can write is a F 0 -measurable random variable satisfying u 0 (x) ≥ 0 for a.e. x ∈ O P-a.s. (A3) Diffusion matrix: a ij ≥ 0 for i = 1, . . . , n, j = 0, . . . , n and there exist π 1 , . . . , π n > 0 such that π i a ij = π j a ji for all i, j = 1, . . . , n (detailed-balance condition).
(A5) Interaction between entropy and noise: There exists C h > 0 such that for all where h is the entropy density defined in (5).
Remark 2 (Discussion of the assumptions). (A1) The Lipschitz regularity of the boundary ∂O is needed to apply the Sobolev and Gagliardo-Nirenberg inequalities. (A2) The regularity condition on u 0 can be weakened to u 0 ∈ L p (Ω; L 2 (O; R n )) for sufficiently large p ≥ 2 (only depending on the space dimension); it is used to derive the higher-order moment estimates. (A3) The detailed-balance condition is also needed in the deterministic case to reveal the entropy structure of the system; see [11]. (A4) The Lipschitz continuity of the stochastic diffusion σ(u) is a standard condition for stochastic PDEs; see, e.g., [36]. (A5) This is the most restrictive assumption. It compensates for the singularity of (∂h/∂u i )(u) = π i log u i at u i = 0. We show in Lemma 33 that satisfies Assumption (A5), where η > 0 and (a k ) ∈ ℓ 2 (R). Taking into account the gradient estimate from the entropy inequality (see (6)), we can allow for more general stochastic diffusion terms like (9); see Lemma 34.

Definition 1 (Martingale solution).
A martingale solution to (1)-(3) is the triple ( U, W , u) such that U = ( Ω, F , P, F) is a stochastic basis with filtration F = ( F t ) t≥0 , W is an ndimensional cylindrical Wiener process, and u = ( u 1 , . . . , u n ) is a continuous D(L) ′ -valued F-adapted process such that u i ≥ 0 a.e. in O × (0, T ) P-a.s., the law of u i (0) is the same as for u 0 i , and for all φ ∈ D(L), t ∈ (0, T ), i = 1, . . . , n, P-a.s., Our main results read as follows. Theorem 4 (Existence for the SKT model without self-diffusion). Let Assumptions (A1)-(A5) be satisfied, let d ≤ 2, and let a 0i > 0 for i = 1, . . . , n. We strengthen Assumption (A4) slightly by assuming that for all v ∈ L 2 (O; R n ), and, for all φ ∈ D(L) ∩ W 2,∞ (O), The weak formulation for the SKT system without self-diffusion is weaker than that one with self-diffusion, since we have only the gradient regularity ∇ u i ∈ L 1 (O), and A ij ( u) may be nonintegrable. However, system (1) can be written in Laplacian form according to (8), which allows for the "very weak" formulation stated in Theorem 4. The condition on γ if d = 2 is needed to prove the fractional time regularity for the approximative solutions.
Remark 5 (Nonnegativity of the solution). The a.s. nonnegativity of the population densities is a consequence of the entropy structure, since the approximate densities u ε i satisfy u ε i = u i (R ε (v ε )) = exp(R ε (v ε )/π i ) > 0 a.e. in Q T . This may be surprising since we do not assume that the noise vanishes at zero, i.e. σ ij (u) = 0 if u i = 0. This condition is replaced by the weaker integrability condition for σ ij (u) log u i in Assumption (A5). A similar, but pointwise condition was imposed in the deterministic case; see Hypothesis (H3) in [25,Section 4.4]. The examples in Section 7 satisfy σ ij (u) = 0 if u i = 0.

Operator setup
In this section, we introduce the operators needed to define the approximate scheme.
3.1. Definition of the connection operator L. We define an operator L that "connects" two Hilbert spaces V and H satisfying V ⊂ H. This abstract operator allows us to define a regularization operator that "lifts" the dual space V ′ to V .
Proposition 6 (Operator L). Let V and H be separable Hilbert spaces such that the embedding V ֒→ H is continuous and dense. Then there exists a bounded, self-adjoint, positive operator L : D(L) → H with domain D(L) = V . Moreover, it holds for L and its dual operator L * : H → V ′ (we identify H and its dual H ′ ) that, for some 0 < c < 1, We abuse slightly the notation by denoting both dual and adjoint operators by A * . The proof is similar to [27,Theorem 1.12]. For the convenience of the reader, we present the full proof.
Proof. We first construct some auxiliary operator by means of the Riesz representation theorem. Let w ∈ H. The mapping V → R, v → (v, w) H , is linear and bounded. Hence, there exists a unique element w ∈ V such that (v, w) V = (v, w) H for all v ∈ V . This defines the linear operator G : H → V , G(w) := w, such that The operator G is bounded and symmetric, since G(w) V = w V = w H and This means that G is self-adjoint as an operator on H.
, G is positive. We claim that G is also one-to-one. Indeed, let G(w) = 0 for some w ∈ H. Then 0 = (v, G(w)) V = (v, w) H for all v ∈ V and, by the density of the embedding V ֒→ H, for all v ∈ H. This implies that w = 0 and shows the claim.
Finally, we define L := Λ 1/2 : D(L) = V → H, which is a positive and self-adjoint operator. Estimate (14) shows that L(v) H = v V for v ∈ V . We deduce from the equivalence between the norm in V and the graph norm of L that, for some which proves the lower bound in (12). The dual operator L * : H → V ′ is bounded too, since it holds for all w ∈ H that This ends the proof.
The following lemma is used in the proof of Proposition 15 to apply Itô's lemma.
Proof. The proof is essentially contained in [27, p. 136ff] and we only sketch it. Let . This shows that every element of D(L −1 ) ′ can be identified with an element of D(L).
Conversely, if w ∈ D(L), we consider functionals of the type v → (v, w) H for v ∈ D(Λ), which are bounded in L −1 (·) H . These functionals can be extended by continuity to functionals F belonging to D(L −1 ) ′ . The proof in [27, p. 137] shows that F D(L −1 ) ′ = w D(L) . We conclude that D(L −1 ) ′ is isometric to D(L). Since Hilbert spaces are reflexive, .
Since u ′′ maps bounded sets to bounded sets, the integral is bounded. Thus, u : For the monotonicity, we use the convexity of h and hence the monotonicity of h ′ : . This proves the lemma.

3.2.
Definition of the regularization operator R ε . First, we define another operator that maps D(L) to D(L) ′ . Its inverse is the desired regularization operator.
where we used the monotonicity of w → u(w) and the lower bound in (12). The coercivity of Q ε is a consequence of the strong monotonicity: . Based on these properties, the invertibility of Q ε now follows from Browder's theorem [20, Theorem 6.1.21].
Lemma 9 shows that the inverse of Q ε exists. We set R ε := Q −1 ε : D(L) ′ → D(L), which is the desired regularization operator. It has the following properties.
. Hence, using (12) and (16), proving that R ε is Lipschitz continuous with Lipschitz constant C/ε. The Fréchet differentiability is a consequence of the inverse function theorem and DR Next, we show that DR ε [v] is Lipschitz continuous. Let w 1 , w 2 ∈ D(L). By Lemma 9, DQ ε [w] is strongly monotone. Thus, for any w ∈ D(L),

Existence of approximate solutions
In the previous section, we have introduced the regularization operator R ε : We clarify the notion of solution to problem (17)- (18). Let T > 0, let τ be an F-adapted stopping time, and let v be a continuous, for a.e. ω ∈ Ω and for all t It can be verified that R ε is strongly measurable and, if v is progressively measurable, also progressively measurable. Furthermore, if w is progressively measurable then so does u(w), ). Therefore, the integrals in (19) are well defined. The local weak solution is called a global weak solution if P(τ = ∞) = 1. Given t > 0 and a process v ∈ L 2 (Ω; C 0 ([0, t]; D(L) ′ )), we introduce the stopping time This time is positive. Indeed, by Chebychev's inequality, it holds for δ > 0 that Then, inserting (19) and using the properties of the operators introduced in Section 3, we can show that P(τ R > δ) ≥ 1 − C(δ), where C(δ) → 0 as δ → 0, which proves the claim. We impose the following general assumptions.
(H1) Entropy density: Let D ⊂ R n be a domain and let h ∈ C 2 (D; [0, ∞)) be such that h ′ : D → R n and h ′′ (u) ∈ R n×n for u ∈ D are invertible and there exists C > 0 such that |u| ≤ C(1 + h(u)) for all u ∈ D. (H2) Initial datum:  We consider general approximate stochastic cross-diffusion systems, since the existence result for (17) may be useful also for other stochastic cross-diffusion systems. Proof. We want to apply Theorem 4.2.4 and Proposition 4.1.4 of [32]. To this end, we need to verify that the operator M : , is Fréchet differentiable and has at most linear growth, DM[v] − cI is negative semidefinite for all v ∈ D(L) ′ and some c > 0, and σ is Lipschitz continuous.
By the regularity of the matrix A and the entropy density h, the operator D(L) → D(L) ′ , w → div(B(w)∇w), is Fréchet differentiable. Then the Fréchet differentiability of R ε (see Lemma 10) and the chain rule imply that the operator M is also Fréchet differentiable with derivative For this, we deduce from the Lipschitz continuity of R ε (Lemma 10) and the property u we obtain for v D(L) ′ ≤ K and ξ ∈ D(L) ′ : Moreover, by Lemma 10 again, It follows from Assumption (A4) and Lemma 8 that These . For the entropy estimate we need two technical lemmas whose proofs are deferred to Appendix A.
We turn to the entropy estimate.
Proposition 15 (Entropy inequality). Let (τ R , v ε ) be a local solution to (17)- (18) and for ω ∈ Ω, t ∈ (0, τ R (ω)). Then there exists a constant C(u 0 , T ) > 0, depending on u 0 and T but not on ε and R, such that Proof. The result follows from Itô's lemma using a regularized entropy. More precisely, we want to apply the Itô lemma in the version of [29,Theorem 3.1]. To this end, we verify the assumptions of that theorem. Basically, we need a twice differentiable function H on a Hilbert space H, whose derivatives satisfy some local growth conditions on H and V , where V is another Hilbert space such that the embedding V ֒→ H is dense and continuous. We choose V = H = D(L) ′ and the regularized entropy Then, in view of the regularity assumptions for h and Lemma 10, H is Fréchet differentiable with derivative where v, ξ ∈ D(L) ′ . In other words, DH[v] can be identified with R ε (v) ∈ D(L). In a similar way, we can prove that DH[v] is Fréchet differentiable with We have, thanks to the Lipschitz continuity of R ε and DR ε [v] (see Lemma 10)  Hence, we can associate DH[v] with L * LR ε (v) ∈ D(L) ′ . Then, by the first estimate in (15) and the Lipschitz continuity of R ε , giving the desired estimate for DH [v] in D(L) ′ . Thus, the assumptions of the Itô lemma, as stated in [29], are satisfied.
To simplify the notation, we set u ε := u(R ε (v R )) and w ε := R ε (v R ) in the following. By Itô's lemma, using DH Lemma 14 shows that the first term on the right-hand side can be estimated from above and integrating by parts, the second term on the right-hand side can be written as The last inequality follows from Assumption (A3), which implies that B(w ε ) = A(u(w ε ))h ′′ (u(w ε )) −1 is positive semidefinite.. We reformulate the last term in (22) by applying Lemma 13 with a = σ(u ε )e k and b = DR ε [v](σ(u ε )e k ): Taking the supremum in (22) over (0, T R ), where T R ≤ T ∧ τ R , and the expectation yields We apply the Burkholder-Davis-Gundy inequality [32, Theorem 6.1.2] to I 1 and use Assumption (A5): Also the remaining integral I 2 can be bounded from above by Assumption (A5): We apply Gronwall's lemma to the function F (t) = sup 0<s<t O h(u ε (s))dx to find that Using this bound in (24) then finishes the proof.
The entropy inequality allows us to extend the local solution to a global one. Then v ε can be extended to a global solution to (19)- (20).
Proof. With the notation u ε = u(R ε (v ε )) and We know from Hypothesis (H1) that |u ε | ≤ C(1 + h(u ε )). Therefore, taking into account the entropy inequality and the second inequality in (15), This allows us to perform the limit R → ∞ and to conclude that we have indeed a solution v ε in (0, T ) for any T > 0.

Proof of Theorem 3
We prove the global existence of martingale solutions to the SKT model with selfdiffusion.

Uniform estimates.
Let v ε be a global solution to (19)- (20) and set u ε = u(R ε (v ε )). We assume that A(u) is given by (3) and that a ii > 0 for i = 1, . . . , n. We start with some uniform estimates, which are a consequence of the entropy inequality in Proposition 15.
Lemma 17 (Uniform estimates). There exists a constant C(u 0 , T ) > 0 such that for all ε > 0 and i, j = 1, . . . , n with i = j, . Moreover, we have the estimate Let v ε be a global solution to (19)- (20). We observe that It is shown in [11,Lemma 4] that for all z ∈ R n and u ∈ (0, ∞) n , Using B(R ε (v ε )) = A(u ε )h ′′ (u ε ) −1 and the previous inequality with z = ∇u ε , we find that Therefore, the entropy inequality in Proposition 15 becomes This is the stochastic analog of the entropy inequality (6). By Hypothesis (H1), we have |u| ≤ C(1 + h(u)) and consequently, which proves (25). Estimate (26) then follows from the Poincaré-Wirtinger inequality. It remains to show estimate (27). We deduce from the second inequality in (15) that This shows that ending the proof.
We also need higher-order moment estimates.
Lemma 18 (Higher-order moments I). Let p ≥ 2. There exists a constant C(p, u 0 , T ), which is independent of ε, such that Moreover, we have Proof. Proceeding as in the proof of Proposition 15 and taking into account identity (22) and inequality (28), we obtain recalling Definition 21 of H(v ε ). We raise this inequality to the pth power, take the expectation, apply the Burkholder-Davis-Gundy inequality (for the second term on the right-hand side), and use Assumption (A5) to find that We neglect the expression ε LR ε (v ε (t)) 2 L 2 (O) and apply Gronwall's lemma. Then, taking into account the fact that the entropy dominates the L 1 (O) norm, thanks to Hypothesis (H1), and applying the Poincaré-Wirtinger inequality, we obtain estimates (30)- (32). Going back to (34), we infer that Combining the previous estimates and arguing as in the proof of Lemma 17, we have This ends the proof.
Using the Gagliardo-Nirenberg inequality, we can derive further estimates. We recall that Q T = O × (0, T ).
Next, we show some bounds for the fractional time derivative of u ε . This result is used to establish the tightness of the laws of (u ε ) in a sub-Polish space. Alternatively, the tightness property can be proved by verifying the Aldous condition; see, e.g., [18]. We recall the definition of the Sobolev-Slobodeckij spaces. Let X be a vector space and let p ≥ 1, α ∈ (0, 1). Then W α,p (0, T ; X) is the set of all functions v ∈ L p (0, T ; X) for which v p W α,p (0,T ;X) = v p L p (0,T ;X) + |v| p W α,p (0,T ;X) With this norm, W α,p (0, T ; X) becomes a Banach space. We need the following technical lemma, which is proved in Appendix A.

5.2.
Tightness of the laws of (u ε ). We show that the laws of (u ε ) are tight in a certain sub-Polish space. For this, we introduce the following spaces: ) with the weak topology T 2 . We define the space , endowed with the topology T that is the maximum of the topologies T 1 and T 2 . The space Z T is a sub-Polish space, since C 0 ([0, T ]; D(L) ′ ) is separable and metrizable and where (v m ) m is a dense subset of L 2 (0, T ; H 1 (O)), is a countable family (f m ) of pointseparating functionals acting on L 2 (0, T ; H 1 (O)). In the following, we choose a number s * ≥ 1 such that Then the embedding H 1 (O) ֒→ L s * (O) is compact.
The set of laws of (u ε ) is tight in with the topology T that is the maximum of T and the topology induced by the L 2 (0, T ; L s * (O)) norm, where s * is given by (40).

5.3.
Convergence of (u ε ). Let P(X) be the space of probability measures on X. We consider the space µ ε W (·) = P(W ∈ ·) ∈ P(C 0 ([0, T ]; U 0 )), recalling the choice (40) of s * . The set of measures (µ ε ) is tight, since the set of laws of (u ε ) and ( √ εL * LR ε (v ε )) are tight in (Z T , T) and (Y T , T Y ), respectively. Moreover, (µ ε W ) consists of one element only and is consequently weakly compact in C 0 ([0, T ]; U 0 ). By Prohorov's theorem, (µ ε W ) is tight. Hence, Z T × Y T × C 0 ([0, T ]; U 0 ) satisfies the assumptions of the Skorokhod-Jakubowski theorem [6,Theorem C.1]. We infer that there exists a subsequence of (u ε , √ εL * LR ε (v ε )), which is not relabeled, a probability space ( Ω, F, P) and, on this By the definition of Z T and Y T , this convergence means P-a.s., We derive some regularity properties for the limit u. We note that u is a Z T -Borel random variable, since We deduce from estimates (25) and (26) and the fact that u ε and u ε have the same law that sup We infer the existence of a further subsequence of ( u ε ) (not relabeled) that is weakly converging in L p ( Ω; L 2 (0, T ; H 1 (O))) and weakly* converging in L p ( Ω; C 0 ([0, T ]; D(L) ′ )) as ε → 0. Because u ε → u in Z T P-a.s., we conclude that the limit function satisfies Let F and F ε be the filtrations generated by ( u, w, W ) and ( u ε , w ε , W ), respectively. By following the arguments of the proof of [7, Proposition B4], we can verify that these new random variables induce actually stochastic processes. The progressive measurability of u ε is a consequence of [4,Appendix B]. Set W ε,k j (t) := ( W ε (t), e k ) U . We claim that W ε,k j (t) for k ∈ N are independent, standard F t -Wiener processes. The adaptedness is a direct consequence of the definition; the independence of W ε,k j (t) and the independence of the increments W ε,k (t) − W ε,k (s) with respect to F s are inherited from (W (t), e k ) U . Passing to the limit ε → 0 in the characteristic function, by using dominated convergence, we find that W (t) are F t -martingales with the correct marginal distributions. We deduce from Lévy's characterization theorem that W (t) is indeed a cylindrical Wiener process.
By definition, u ε i = u i (R ε (v ε )) = exp(R ε (v ε )) is positive in Q T a.s. We claim that also u i is nonnegative in O a.s. Lemma 24 (Nonnegativity). It holds that u i ≥ 0 a.e. in Q T P-a.s. for all i = 1, . . . , n.
Proof. Let i ∈ {1, . . . , n}. Since u ε i > 0 in Q T a.s., we have E (u ε i ) − L 2 (0,T ;L 2 (O)) = 0, where z − = min{0, z}. The function u ε i is Z T -Borel measurable and so does its negative part. Therefore, using the equivalence of the laws of u ε i and u ε i in Z T and writing µ ε i and µ ε i for the laws of u ε i and u ε i , respectively, we obtain This shows that u ε i ≥ 0 a.e. in Q T P-a.s. The convergence (up to a subsequence) u ε → u a.e. in Q T P-a.s. then implies that u i ≥ 0 in Q T P-a.s.
The following lemma is needed to verify that ( u, W ) is a martingale solution to (1)-(2).
To show that the limit is indeed a solution, we define, for t ∈ [0, T ], i = 1, . . . , n, and φ ∈ D(L), The following corollary is a consequence of the previous lemma.
Corollary 26. It holds for any φ 1 ∈ L 2 (O) and φ 2 ∈ D(L) that Since v ε is a strong solution to (17), it satisfies for a.e. t ∈ [0, T ] P-a.s., i = 1, . . . , n, and φ ∈ D(L), We deduce from the equivalence of the laws of (u ε , εL By Corollary 26, we can pass to the limit ε → 0 to obtain This identity holds for all i = 1, . . . , n and all φ ∈ D(L). This shows that We infer from the definition of Λ i that

Proof of Theorem 4
We turn to the existence proof of the SKT model without self-diffusion.
The following lemma is needed to derive the fractional time estimate.

Proof. The Hölder inequality and (30) immediately yield
≤ C, and we conclude from the Poincaré-Wirtinger inequality, estimate (32), and the previous estimate that . Taking the expectation and applying the Hölder inequality, we infer that The bounds (52)-(53) yield, after taking the expectation and applying Hölder's inequality again, the conclusion (51).
We show now that the fractional time derivative of u ε is uniformly bounded.
Lemma 29 (Fractional time regularity). Let d ≤ 2. Then there exist 0 < α < 1, p > 1, and β > 0 such that αp > 1 and We proceed similarly as in the proof of Lemma 21. First, we estimate the diffusion part, setting dr.
(The result holds for any space dimension if γ < 2/d.) Taking into account (33) is bounded in the latter space. We turn to the estimate of u ε in the W α,p (0, T ; D(L) ′ ) norm: . It remains to consider the last term. In view of estimate (15) and the Lipschitz continuity of R ε with Lipschitz constant C/ε, we obtain Moreover, by (15) and the Lipschitz continuity of R ε again, where we used estimate (33). This finishes the proof.
6.2. Tightness of the laws of (u ε ). The tightness is shown in a different sub-Polish space than in Section 5.2: , endowed with the topology T that is the maximum of the topology of C 0 ([0, T ]; D(L) ′ ) and the weak topology of L ρ 1 w (0, T ; W 1,ρ 1 (O)), recalling that ρ 1 = (d + 2)/(d + 1) > 1. Lemma 30. The family of laws of (u ε ) is tight in with the topology that is the maximum of T and the topology induced by the L 2 (0, T ; L 2 (O)) norm.
Proof. The tightness in L 2 (0, T ; L q (O)) for q < d/(d − 1) = 2 is a consequence of the compact embedding W 1,1 (O) ֒→ L q (O) as well as estimates (47) and (54). In fact, we can extend this result up to q = 2 because of the uniform bound of u ε i log u ε i in L ∞ (0, T ; L 1 (O)), which originates from the entropy estimate. Indeed, we just apply [3, Prop. 1], using additionally (26) with a i0 > 0. Then the tightness in L 2 (0, T ; L 2 (O)) follows from Lemma 36. Finally, the tightness in Z T is shown as in the proof of Lemma 22 in Appendix B.
In three space dimensions, we do not obtain tightness in L 2 (0, T ; L 2 (O)) but in the larger space L 4/3 (0, T ; L 2 (O)). This follows similarly as in the proof of Lemma 22 taking into account the compact embedding W 1,ρ 1 (O) ֒→ L 2 (O), which holds as long as d ≤ 3, as well as estimates (50) and (54). Unfortunately, this result seems to be not sufficient to identify the limit of the product u ε i u ε j . Therefore, we restrict ourselves to the two-dimensional case. The following result is shown exactly as in Lemma 23.
Arguing as in Section 5.3, the Skorokhod-Jakubowski theorem implies the existence of a subsequence, a probability space ( Ω, F, P), and, on this space, (Z T × Y T × C 0 ([0, T ]; U 0 ))valued random variables ( u ε , w ε , W ε ) and ( u, w, W ) such that ( u ε , w ε , W ε ) has the same law as (u ε , ) and, as ε → 0 and P-a.s., This convergence means that P-a.s., The remainder of the proof is very similar to that one of Section 5.3, using slightly weaker convergence results. The most difficult part is the convergence of the nonlinear term ∇( u ε i u ε j ), since the previous convergences do not allow us to perform the limit u ε i ∇ u ε j because of ρ 1 < 2. The idea is to consider the "very weak" formulation by performing the limit in u ε i u ε j ∆φ instead of ∇( u ε i u ε j ) · ∇φ for suitable test functions φ. Indeed, let φ ∈ L ∞ (0, T ; C ∞ 0 (O)). Since u ε i → u strongly in L 2 (0, T ; L 2 (O)) P-a.s., we have It follows from the equivalence of the laws that and we conclude from Vitali's theorem that By density, this convergence holds for all test functions φ ∈ L ∞ (0, T ; W 2,∞ (O)) such that ∇φ · ν = 0 on ∂O. This ends the proof of Theorem 4.
Remark 32 (Three space dimensions). The three-dimensional case is delicate since u ε i lies in a space larger than L 2 (Q T ). We may exploit the regularity (51) for ∇(u ε i u ε j ), but this leads only to the existence of random variables η ε ij and η ij with i, j = 1, . . . , n and i = j on the space X T = L ρ 2 w (0, T ; L ρ 2 (O)) such that η ε ij and u ε i u ε j have the same law on B(X T ) and, as ε → 0, η ε ij ⇀ η ij weakly in X T . Similar arguments as before lead to the limit but we cannot easily identify η ij with u i u j .

Discussion of the noise terms
We present some examples of admissible terms σ(u). Recall that (e k ) k∈N is an orthonormal basis of U. Proof. With the entropy density h given by (5), we compute (∂h/∂u i )(u) = π i log u i and (∂ 2 h/∂u i ∂u j )(u) = (π i /u i )δ ij . Therefore, by Jensen's inequality and the elementary inequalities |u i log u i | ≤ C(1 + u 1+η i ) for any η > 0 and |u| ≤ C(1 + h(u)), The second condition in Assumption (A5) becomes Thus, Assumption (A5) is satisfied.
The proof shows that J 1 can be estimated if s(u i ) 2 log(u i ) 2 is bounded from above by C(1 + h(u)). This is the case if s(u i ) behaves like u α i with α < 1/2. Furthermore, J 2 can be estimated if s(u i ) 2 /u i is bounded, which is possible if s(u i ) = u α i with α ≥ 1/2. Thus, to both satisfy the growth restriction and avoid the singularity at u i = 0, we have chosen σ ij as in Lemma 33. This example is rather artificial. To include more general choices, we generalize our approach. In fact, it is sufficient to estimate the integrals in inequality (23) in such a way that the entropy inequality of Proposition 15 holds. The idea is to exploit the gradient bound for u i for the estimatation of J 1 and J 2 .
Consider a trace-class, positive, and symmetric operator Q on L 2 (O) and the space U = Q 1/2 (L 2 (O)), equipped with the norm Q 1/2 (·) L 2 (O) . We will work in the following with an U-cylindrical Wiener process W Q . This setting is equivalent to a spatially colored noise on L 2 (O) in the form of a Q-Wiener process (with Q = Id). The latter viewpoint provides, in our opinion, a more intuitive insight. In particular, the operator Q is constructed from the eigenfunctions and eigenvalues described below.
Let (η k ) k∈N be a basis of L 2 (O), consisting of the normalized eigenfunctions of the Laplacian subject to Neumann boundary conditions with eigenvalues λ k ≥ 0, and set Considering a sequence of independent Brownian motions (W k 1 , . . . , W k n ) k∈N , we assume the noise to be of the form a k e k W k j (t), j = 1, . . . , n, t > 0, and (e k ) k∈N = (a k η k ) k∈N is a basis of U = Q 1/2 (L 2 (O)).
Proof. We can write inequality (23) for 0 < T < T R as recalling that w ε = R ε (v ε ) and u ε = u(w ε ). We simplify J 3 and J 4 , using the definition e k = a k η k : The last inequality follows from our assumption on (a k ). By (28), we can estimate the integrand of the third integral on the left-hand side of (55) according to Hence, because of |u| ≤ C(1 + h(u)), we can formulate (55) as It is sufficient to continue with the case α = 1, since the proof for α < 1 follows from the case α = 1. Then, using |u ε | ≤ C(1 + h(u ε )), Now, we use the following lemma which is proved in Appendix A.
For sufficiently small δ > 0, the last terms on the right-hand side can be absorbed by the corresponding terms on the left-hand side, leading to Gronwall's lemma ends the proof.
In the case without self-diffusion, we have an H 1 (O) estimate for (u ε i ) 1/2 only, and it can be seen that stochastic diffusion terms of the type δ ij u α i for α > 1/2 are not admissible. However, we may choose σ ij (u)e k = δ ij u α i (1 + (u ε i ) β ) −1 a k η k for 1/2 ≤ α < 1 and β ≥ α/2. The matrix u ′ (w) = (h ′′ ) −1 (u(w)) is symmetric and positive semidefinite (since h is convex). Thus, the square root operator u ′ (w) exists and is symmetric. This shows that Inserting these relations into (57) leads to and consequently, Together with (58) we obtain the statement.

We find after an integration that
which yields the statement. observing that lim t→s (t − s) 1−δ t s g(r)dr = 0 for 1 − δ > −1, since the integrability of g implies that lim t→s (t − s) −1 t s g(r)dr = g(s) for a.e. s. The result follows as the integrals on the right-hand side of (59) are finite.

Appendix B. Tightness criterion
Lemma 36 (Tightness criterion). Let O ⊂ R d (d ≥ 1) be a bounded domain with Lipschitz boundary and let T > 0, p, q, r ≥ 1, α ∈ (0, 1) if r ≥ p and α ∈ (1/r − 1/p, 1) if r < p. Let s ≥ 1 be such that the embedding W 1,q (O) ֒→ L s (O) is compact, and let Y be a Banach space such that the embedding L s (O) ֒→ Y is continuous. Furthermore, let (u n ) n∈N be a sequence of functions such that there exists C > 0 such that for all n ∈ N, E u n L p (0,T ;W 1,q (O)) + E u n W α,r (0,T ;Y )) ≤ C.
Then the laws of (u n ) are tight in L p (0, T ; L s (O)) if q ≤ d and in L p (0, T ; C 0 (O)) if q > d. If p = ∞, the space L p (0, T ; ·) is replaced by C 0 ([0, T ]; ·).
The definition of tightness finishes the proof.

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