The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.


Introduction
Pattern formation at the ecosystem level has recently gained a lot of attention in spatial ecology and its mathematical modeling. Theoretical models are a widely used tool for studying e.g. banded vegetation patterns. One important model is the system of advection-diffusion equations proposed by Klausmeier [41]. This model for vegetation dynamics in semi-deserted areas is based on the "water redistribution hypothesis", using the idea that rain water in dry regions is eventually infiltrated into the ground. Water falling down onto bare ground mostly runs off downhill toward the next patch of vegetation which provides a better infiltration capacity. The soil in such regions of the world as Australia, Africa, and Southwestern North America is prone to nonlocality of water uptake due to the semi-arid environment. Studies of the properties of the system and further developments can be found in e.g. [40,[61][62][63][64]71].
The Klausmeier system is a generalization of the so-called Gray-Scott system [31] (see also [60,65] for earlier accounts employing similar models) which already exhibits effects similar to Turing patterns [20,23,50,70], see for instance the discussion in [72]. We refer to [45,52,53] for further reading on pattern formation in biology. A discussion of reaction-diffusion type equations with motivation from biology can be found in [56].
The underlying mathematics of this model is given by a pair of solutions (u, v) to a partial differential equation system coupled by a nonlinearity. The function u represents the surface water content and v represents the biomass density of the plants. In order to model the spread of water on a terrain without a specific preference for the direction in which the water flows, the original models were extended by replacing the diffusion operator by a nonlinear porous media operator, which represents the situation that the ground is partially filled by interconnected pores conveying fluid under an applied pressure gradient.
The deterministic or macroscopic model is derived from the limiting behavior of interacting diffusions -the so-called microscopic model, see [43]. When applying the strong law of large numbers and passing from the microscopic to the macroscopic equation, one is neglecting the random fluctuations. In order to get a more realistic model, it is necessary to add noise, which represents the randomness of the natural environment or the fluctuation of parameters in the model. The introduction to stochasticity to ecological models is supported by the arguments of [33]. Due to the Wong-Zakai principle, this leads to the representation of the noise as a Gaussian stochastic perturbation with Stratonovich increment, see [27,Section 3.4] and [49,78]. One important consequence is the preservation of energy in the noisy system.
In practice, we are investigating the system (1.1) driven by a multiplicative infinite dimensional Wiener process. Under suitable regularity assumptions on the initial data (u 0 , v 0 ), on γ and on the perturbation by noise which we shall specify later, we find that there exists a nonnegative (martingale) solution to the system (1.2) in dimensions d = 1, 2, 3, see our main result Theorem 3.6. More precisely, we prove existence of nonnegative solutions to the noisy systems (3.6)-(3.7) (Itô noise), (3.1)-(3.2) (Stratonovich noise) respectively. For d = 1, we show that the solution to (3.6)-(3.7) is pathwise unique and that there exists a strong solution in the stochastic sense. Thus, we are actually considering the following system for γ > 1 Here, Ξ 1 and Ξ 2 denote independent random Gaussian noises specified to be certain Banach space valued linear Stratonovich Wiener noises later on. Similar stochastic equations with Lévy noise have been studied e.g. in [10,18]. We note that due to the coupling and the nonlinearity, the solutions are not expected to be stochastically independent.
Nonlinear diffusion systems perturbed by noise exhibit certain improved well-posedness and regularity properties when compared to their deterministic counterparts, see, among others, [4,15,27]. Another feature of linear multiplicative noise is that it preserves the nonnegativity of the initial condition [1,68]. Some applications may demand more complicated noises with nonlinear structure or coupling, however, we choose a linear dependence on the noise as a first step toward more general models.
The challenging problem in the system given by (1.1), respective the noisy system (1.2), is the nonlinearity appearing once with a negative sign and once with a positive sign. Choosing different speeds of diffusion seems rather natural, as it well known that the characteristic pattern formation may not take place if r u = r v , cf. [36]. The stochastic perturbation, however, does not restrict pattern formation [12,79], noting that our choice of noise exhibits rather small intensity and damped high frequency modes. On the other hand, the nonlinearity is not of variational structure, such that energy methods are not available for the analysis, and neither the maximum principle nor Gronwall type arguments work. More direct deterministic (pathwise) methods may also fail in general as the equation for v is non-monotone.
Another difficulty is posed by the nonlinear porous media diffusion operator in the equation for u. Typically, it is studied with the help of variational monotonicity methods, see e.g. [2,48]. To the nonlinear diffusion, neither semigroup methods nor Galerkin approximation can be applied directly without greater effort. Other approaches are given for instance in [26,29,55,74], to mention a few.
Our motivation to prove a probabilistic Schauder-Tychonoff type fixed point theorem originates from the aim to show existence of a solution to the stochastic counterpart (1.2) of the system (1.1). Our result is for instance also applied in [34,35]. As standard methods for showing existence and uniqueness of solutions to stochastic partial differential equations cannot be applied directly, we perform a fixed point iteration using a mix of so-called variational and semigroup methods with nonlinear perturbation. Here, a precise analysis of regularity properties of nonnegative solutions to regularized and localized subsystems with truncated nonlinearities needs to be conducted. Together with the continuous dependence of each subsystem on the other one, we obtain the fixed point, locally in time, by weak compactness and an appropriate choice of energy spaces. The result is completed by a stopping time gluing argument.
Apart from the probabilistic structure, the main novelty is that we construct the fixed point in the nonlinearity and not in the noise coefficient. This works well because our system is coupled precisely in the nonlinearity. The main task consists then in the analysis of regularity and invariance properties of a regularized system with "frozen" nonlinearity (not being fully linearized, however, as the porous media operator remains).
It may be possible to apply this method in the future to other nonlinear systems, as for example, systems with nonlinear convection-terms as systems with transport or Navier-Stokes type systems. It is certainly possible to apply the method to linear cross diffusion systems. See [19] for a recent work proving existence of martingale solutions to stochastic cross diffusion systems, however, their approach relies on other methods that do not cover the porous media case. See [51] for a previous work using the classical Schauder theorem for stochastic evolution equations with fractional Gaussian noise.
Given higher regularity of the initial data, we also show the pathwise uniqueness of the solution to (1.2) for spatial dimension d = 1. As a consequence of the celebrated result by Yamada and Watanabe [75,80], we obtain existence and uniqueness of a strong solution, see Corollary 3.12. We refer to [11,13,14,59] for previous works that employ a similar strategy for proving the existence of a unique strong solution.
Structure of the paper. The stochastic Schauder-Tychonoff type Theorem 2.1 is presented in Section 2. In the subsequent section, i.e. Section 3, we apply this fixed point theorem to show the existence of a martingale solution to the stochastic counterpart (1.2) of the system (1.1). Section 3 contains our main result Theorem 3.6. Section 4 is devoted to the proof of Theorem 2.1. In Section 5, we prove several technical propositions that are need for the main result in Section 3. Section 6 contains the proof of pathwise uniqueness, that is, Theorem 3.10. Some auxiliary results are collected in Appendices A, B and C. } be a reflexive Banach function space embedded compactly and densely into X. In both cases, the trajectories take values in a Banach function space E over the spatial domain O, where we assume that E has the UMD property, see [73]. Let A = (Ω, F, F, P) be a filtered probability space with filtration F = (F t ) t∈[0,T ] satisfying the usual conditions. Let H be a separable Hilbert space and (W (t)) t∈[0,T ] be a Wiener process 2 in H with a linear, nonnegative definite, symmetric trace class covariance operator Q : H → H such that W has the representation where {ψ i : i ∈ I} is a complete orthonormal system in H, I a suitably chosen countable index set, and {β i : i ∈ I} a family of independent real-valued standard Brownian 1 Here, D ′ (O) denotes the space of Schwartz distributions on O, that is, the topological dual space of smooth functions with compact support D(O) = C ∞ 0 (O). 2 That is, a Q-Wiener process, see e.g. [16] for this notion.
For m ≥ 1, define the collection of processes ξ is F-progressively measurable and E|ξ| m X < ∞ equipped with the semi-norm where w is the solution to the following Itô stochastic partial differential equation (SPDE) For convenience, we drop the dependence on the initial datum w 0 in the notation V(ξ).
Here, we implicitly assume that (2.2) is well-posed and a unique strong solution (in the stochastic sense) w ∈ M m A (X) exists for ξ ∈ M m A (X). Here, we shall also assume that A : D(A) ⊂ E → E is a possibly nonlinear and measurable (single-valued) operator and and assume that Σ : E → γ(H, E) is strongly measurable. Here, γ(H, E) denotes the space of γ-radonifying operators from H to E, as defined in the beginning of Section 5, which coincides with the space of Hilbert-Schmidt operators L HS (H, E) if E is a separable Hilbert space.
We are ready to formulate our main tool for proving the existence of nonnegative martingale solutions to (1.2), that is, a stochastic variant of the (deterministic) Schauder-Tychonoff fixed point theorem(s) from [30, § 6-7].
Theorem 2.1. Let H be a Hilbert space, Q : H → H such that Q is linear, symmetric, nonnegative definite and of trace class, let U be a Banach space, and let us assume that we have a compact and dense embedding X ′ → X as above. Let m > 1. Suppose that for any filtered probability space A = (Ω, F, F, P) and for any Q-Wiener process W with values in H that is modeled on A the following holds.
Suppose that there exist constants R 1 , . . . , R K > 0, K ∈ N, continuous functions and a nonempty, sequentially weak * -closed, measurable and bounded subset 3 Let us assume that the operator V A,W , defined by (2.2), restricted to X R 1 ,...,R K (A) satisfies the following properties: Then, there exists a filtered probability space A * = (Ω * , F * , F * , P * ) (that satisfies the usual conditions) together with a Q-Wiener process W * modeled on A * and an element . The proof of Theorem 2.1 is postponed to Section 4. We note that by construction, we get that w * solves dw * (t) = (Aw * (t) + F (w * , t)) dt + Σ(w * (t)) dW * (t), w * (0) = w 0 . on A * .

Existence of a solution to the stochastic Klausmeier system
In this section, we shall prove the existence of a nonnegative solution to the stochastic Klausmeier system. First, we will introduce some notation and the definition of a (martingale) solution. After fixing the function spaces and the main hypotheses on the parameters of those, we present our main result Theorem 3.6. The following proof consists mainly of a verification of the conditions for Theorem 2.1 and a stopping time localization procedure. As pointed out before, we are using compactness arguments to show the existence, which leads to the loss of the initial stochastic basis.
Let H 1 and H 2 be a pair of separable Hilbert spaces, let A = (Ω, F, F, P) be a filtered probability space and let W 1 , W 2 be a pair of independent Wiener process modeled on A taking values in H 1 and H 2 , respectively, with covariance operators Q 1 and Q 2 , 4 Note that we can relax assumption (iv) to m0 > 1 if we assume additionally that there exist respectively. We are interested in the solution to the following reduced Klausmeier system for x ∈ O and t > 0, and, with Neumann (or periodic if O = [0, 1] d ) boundary conditions and initial conditions u(0) = u 0 and v(0) = v 0 . Let r u , r v , χ > 0 be positive constants. Here, we use the abbreviation x [γ] := |x| γ−1 x for γ > 1. The hypotheses on the linear noise coefficient maps σ 1 , σ 2 are specified below. Due to the nonlinear porous media term, we do neither use solutions in the strong stochastic sense, nor mild solutions, that is, solutions in the sense of stochastic convolutions.
Let us define what we mean with a solution on a fixed stochastic basis. The function spaces H −1 2 (O) and H ρ 2 (O) used in the following definition are discussed in Appendix B.
such that u and v are F = (F t ) t∈[0,T ] -adapted, and satisfy for every t ∈ [0, T ], P-a.s. in H ρ 2 (O). Here, V := L γ+1 (O) is dualized in a Gelfand triple over H := H −1 2 (O) which, in turn, is identified with its own dual by the Riesz isometry. Thus V * is not to be mistaken to be equal to be the usual Banach space dual of V , namely L γ+1 γ (O), see the proof of Theorem 5.4 for details. Note that the above ds-integral for u in (3.3) is initially a V *valued Bochner integral, however seen to be in fact H −1 2 (O)-valued, see the discussion in [48, Section 4.2] for further details.
As mentioned before, due to the loss of the original probability space, we are considering solutions in the weak probabilistic sense.
such that (i) the quadruple A := (Ω, F, F, P) is a complete filtered probability space with a filtration F = (F t ) t∈[0,T ] satisfying the usual conditions, (ii) W 1 and W 2 are independent H 1 -valued, respectively, H 2 -valued Wiener processes over the probability space A with covariance operators Q 1 and Q 2 , respectively, are F-adapted processes such that the couple (u, v) is a solution to the system (3.1) and (3.2) over the probability space A in the sense of Definition 3.1 for some ρ ∈ R. Remark 3.3. For our purposes, instead of the Stratonovich formulation the system (3.1)-(3.2), it is convenient to consider the equations in Itô form: and, with initial data u(0) = u 0 and v(0) = v 0 . One reason for this is that the stochastic integral then becomes a local martingale. In order to show the existence of a solution to the Stratonovich system, one would have to incorporate the Itô-Stratonovich conversion term (cf. [25]), which, due to the linear multiplicative noise, is a linear term being just a scalar multiple of u, v, respectively. If one is interested in the exact form of the correction term, we refer to [21]. We also refer to the discussion in [36], where the constant accounting for the correction term is computed explicitly.
From now on, we shall consider the system (3.1)-(3.2) in Itô form, that is, as in (3.6)-(3.7). Definitions 3.1 and 3.2 are supposed to be adjusted in the obvious way, keeping the statement on the regularity of the solution unchanged. We note that we can solve (3.1)-(3.2) by a straightforward modification of the proof given here.
Before presenting our main result, we will first introduce the hypotheses on d, γ, ρ and the initial conditions u 0 and v 0 , and on the multiplication operators σ 1 , σ 2 . Most of the hypotheses are technical in nature, as they lead to several different embeddings for function spaces and interpolation spaces that we need to use in our proofs.
Let us assume that the initial conditions u 0 , v 0 are F 0 measurable and satisfy and that u 0 and v 0 are a.e. nonnegative functions (nonnegative Borel measures that are finite on compact subsets, respectively).
Hypothesis 3.5 (Noise). Let {ψ k : k ∈ Z} be the eigenfunctions of −∆ and {ν k : k ∈ Z} the corresponding eigenvalues. Let W 1 and W 2 be a pair of Wiener processes given by : k ∈ Z}, j = 1, 2, is a pair of nonnegative sequences belonging to ℓ 2 (Z), {β 2}} is a family of independent standard real-valued Brownian motions. We assume that λ Compare also with Subsection 5.1 for details.
Under these hypotheses, the existence of a martingale solution can be shown.
(iii) for any choice of parameters ρ, m 0 , l as in Hypothesis 3.4, there exists a constant C 2 (T ) > 0 such that We shall also collect a standard notion on uniqueness.
Definition 3.7. The system (3.6)-(3.7) is called pathwise unique if, whenever (u i , v i ), i = 1, 2 are martingale solutions to the system (3.6)-(3.7) on a common stochastic basis (Ω, F, F, P, (W 1 , W 2 )), such that P(u 1 (0) = u 2 (0)) = 1 and P(v 1 (0) = v 2 (0)) = 1, then The theorem of Yamada and Watanabe [80] asserts that (weak) existence and pathwise uniqueness of the solution to a stochastic equation is equivalent to the existence of a unique strong solution. Therefore, showing pathwise uniqueness is a fundamental step for obtaining the existence of a unique strong solution. Under certain additional conditions, as collected below, we are able to prove pathwise uniqueness. Remark 3.9. The condition δ 0 > 0 implies that the Hypothesis 3.8 can only be satisfied if d = 1.
Proof. The proof is postponed to Section 6. □ By this Theorem at hand, the following corollary is a consequence of the Theorem of Yamada-Watanabe. Proof. This follows from Theorem 3.6 in combination with Theorem 3.10 and the Yamada-Watanabe theorem, see [48,Appendix E] and [44,54,57]. □ The proof of Theorem 3.6 is an application of the Schauder-Tychonoff-type Theorem 2.1 and consists of the following five steps to verify the conditions of Theorem 2.1.
In the first step, we are specifying the underlying Banach spaces. In the second step, we shall construct the operator V for a truncated system and show that the operator V satisfies the assumptions of Theorem 2.1. In the third step, we localize via stopping times and glue the fixed point solutions together, which exist by Theorem 2.1. In the fourth step, we prove that the stopping times are uniformly bounded. In the fifth step, we show that we indeed yield a martingale solution satisfying the above properties. However, to keep the proof itself simple, we will postpone several technical a priori estimates and further regularity results which are collected in Section 5.
Proof of Theorem 3.6.
Step (I). The underlying space(s). Here we define the spaces on which the operator V will act. Let the probability space A = (Ω, F, F, P) be given and let W 1 and W 2 be two independent H 1 and H 2 -valued Wiener processes defined over A with covariances Q 1 and Q 2 .
Let us define the Banach space equipped with the norm ∥η∥ Y := ∥η∥ L γ+1 (0,T ;L γ+1 ) + ∥η∥ L ∞ (0,T ;H −1 2 ) , η ∈ Y, and the reflexive Banach space Finally, let us fix an auxiliary Banach space then one can show by Sobolev embedding and interpolation theorems (see Proposition A.6 in the appendix) that there exists a constant C > 0 such that Let us fix the compactly embedded reflexive Banach subspace of Z by 6 Note that here, progressive measurability is meant relative to the Borel σ-fields of the target spaces H −1 2 (O) and H ρ 2 (O) respectively. Finally, for fixed R 0 , R 1 , R 2 > 0, let us define the subspace X A = X A (R 0 , R 1 , R 2 ) by 6 For the definition of W α m 0 , we refer to Appendix B.
It is easy to verify that X A is a sequentially weak * -closed and bounded subset of The continuous functions Ψ i , i = 0, 1, 2, satisfying assumption (a) of Theorem 2.1 can be defined in the obvious way. Also, it is easy to find measurable functions Θ i , i = 0, 1, 2, with closed sublevel sets such that assumption (b) of Theorem 2.1 is satisfied, and can be used to capture the nonnegativity by setting e.g. Θ i ((η, ξ)) := ∞ if η or ξ is negative, i = 0, 1, 2.
Every (η, ξ) ∈ X A is a pair of a (equivalence class of a) nonnegative function (or a nonnegative Borel measure that is finite on compact subsets of O).
Remark 3.14. Note that in order to apply Theorem 2.1 formally, we will assume the obvious modification (or extension) of its statement and proof, such that we can treat pairs of spaces with different exponents like M 2,m 0 A (Y, Z).
Step (II). The truncated system. Let ϕ ∈ D(R) be a smooth cutoff function that satisfies and let ϕ κ (x) := ϕ(x/κ), x ∈ R, κ ∈ N. In addition, for any progressively measurable pair of processes where ν ∈ (0, 1] is chosen such that 1 Let us consider the truncated system given by (3.10) du We shall show the existence of a martingale solution to system (3.10)-(3.11).
Proposition 3.15. For any κ ∈ N, there exists constants R 0 > 0, R 1 > 0, R 2 > 0, depending on κ, such that there exists a martingale solution (u κ , v κ ) to system (3.10)- Proof of Proposition 3.15. The proof consists of several steps. First, we shall define an operator denoted by V κ which satisfies the assumptions of Theorem 2.1, yielding the existence of a martingale solution.
Step (a) Definition of the operator V κ . First, define and v κ is a solution to In fact, by Theorem 5.4 and Proposition 5.5, given such a pair of processes (η, ξ), the existence of a nonnegative unique solution u κ to (3.12) for nonnegative initial data u 0 with follows. By Proposition 5.7 and Proposition 5.8, the existence of a unique solution v κ to (3.13) with

follows.
Step (5.6). Then, due to Proposition 5.6, we know that Furthermore, by Proposition 5.7, the existence of a unique solution to (3.13) such that . Finally, by Proposition 5.9 we can show that for Step (c) Continuity of V κ on X . Next we show that assumption (ii) of Theorem 2.1 is satisfied. We need to show that the restriction of the operator Firstly, due to Proposition 5.12 and Remark 3.13, we know that the sequence {u Secondly, due to Proposition 5.11 and keeping Remark 3.13 in mind, it follows that the sequence {v (3.13). This gives that the operator is continuous.
Step (d) Tightness. Next we show that assumption (iii) of Theorem 2.1 is satisfied. First, note that the embedding Z ′ → Z is compact by the Aubin-Lions-Simon Lemma B.1. Second, by Proposition 5.9 it follows that for any κ ∈ N there exists a constant C = Hence, by Proposition 5.13 and by standard arguments (cf. [24]), we know that the laws of the set

By Proposition 5.13 and the Aubin-Lions-Simon Lemma B.1, the laws of {u
Step (e) Continuity of paths. To show assumption (iv) of Theorem 2.1, we notice that by choosing , it follows from Proposition 5.13 and standard arguments on continuity of solutions to nonlinear SPDEs that Step (f ) Existence of a fixed point.

□
Step (III). Here, we will construct a family of solutions {(ū κ ,v κ ) : κ ∈ N} following the solution to the original problem until a stopping timeτ κ . In particular, we will introduce for each κ ∈ N a new pair of processes (ū κ ,v κ ) following the Klausmeier system up to the stopping timeτ κ . Besides, we will have Let us start with κ = 1. From Proposition 3.15, we know there exists a martingale solution consisting of a probability space A 1 = (Ω 1 , F 1 , F 1 , P 1 ), two independent Wiener processes (W 1 1 , W 1 2 ) defined over A 1 , and a couple of processes (u 1 , v 1 ) solving P 1 -a.s.
Let us define now the stopping time , the pair (u 1 , v 1 ) solves the system given in (1.2). Now, we define a new pair of processes (ū 1 ,v 1 ) following (u 1 , v 1 ) on [0, τ * 1 ) and extend this processes to the whole interval [0, T ] in the following way. First, we put A 1 := A 1 andW 1 j := W 1 j , j = 1, 2, and let us introduce the processes y 1 and y 2 being a strong solution over A 1 to can be shown by standard methods, cf. [16]. Furthermore, it is straightforward to verify that the assumptions on the initial conditions are satisfied. Now, let us define two processesū 1 andv 1 being identical to u 1 and v 1 , respectively, on the time interval [0, τ * 1 ) and following the porous media, respective, the heat equation (with lower order terms) with noise and without nonlinearity, i.e., Let us now construct the probability space and the processes for the next time interval.
. Again, from Proposition 3.15, we know there is a martingale solution consisting of a probability space A 2 = (Ω 2 , F 2 , F 2 , P 2 ), a pair of independent Wiener processes (W 2 1 , W 2 2 ) such that (W 1 1 ,W 1 2 , W 2 1 , W 2 2 ) are independent as well, a couple of processes (u 2 , v 2 ) solving P 2 -a.s. the system with initial condition (u 2 (0), v 2 (0)) having law µ 1 . Let us define now the stopping times on A 2 , . Finally, let us set for j = 1, 2 which give independent Wiener processes for j = 1, 2, w.r.t. the filtration F 2 . Now, let us define two processesū 2 andv 2 being identical toū 1 andv 1 , respectively, on the time interval [0, τ * 1 ), being identical to u 2 and v 2 on the time interval [τ * 1 , τ * 1 + τ * 2 ) and following the porous media, respective, the heat equation (with lower order terms) with multiplicative noise, afterwards. Let us note, for any initial condition having distribution equal to u 2 (τ * 2 ) and v 2 (τ * 2 ) that there exists a strong solutions y 1 (·, ·, τ * 2 + τ * 1 ) and y 2 (·, ·, τ * 2 + τ * 1 ) of the systems (3.19) and (3.20), respectively, on In the same way we will construct for any κ ∈ N a probability space A κ , a pair of independent Wiener processes (W 1 κ ,W 2 κ ), over A κ and a pair of processes (ū κ ,v κ ) starting at (u 0 , v 0 ) and solving system (3.10)-(3.11) up to timē and following the porous media, respective, the heat equation afterwards. Besides, we know that Step (IV). Uniform bounds on the stopping time. Let us consider the family {(ū κ ,v κ ) : κ ∈ N}. The next aim is to show that there exists First, that due to Proposition 5.14 there exists a constant C 0 (1, T ) > 0 such that From above we can conclude that there exists a constant C > 0 such that Here, it is important thatū κ ≥ 0. This estimate can be extended to the time interval [0, T ] by standard results (see e.g. [2] and Proposition 5.14). Next, let us assume R 1 > 0 is that large that where the constants C(T ), δ 0 , δ 1 are given in Proposition 5.15. Observe, by Proposition 5.15, we have for l as in Hypothesis 3.4 Then, by Proposition 5.14, we know that for p = 1, the term which can be uniformly bounded by R l 0 . Hence, we conclude by (3.23 Due to the choice of m and m 0 , we know that the inequality is satisfied. Hence, setting In particular, there exists R 0 , R 1 and R 2 such that for all κ ∈ N we have Step (V). Passing on to the limit. In the final step, we will show that P-a.s. a martingale solution to (3.6)-(3.7) exists.
There exists a measurable set Ω 0 ⊂ Ω with P(Ω 0 ) = 1 such that a martingale solution Proof. For any κ ∈ N, let us define the set It can be clearly observed that there exists a progressively measurable process (u, v) over A such that (u, v) solves P-a.s. the integral equation given by (3.6)-(3.7) up to time T .
In particular, we have for the conditional probability thus by the Markov inequality, The proof of Theorem 3.6 is complete. □

Proof of the stochastic Schauder-Tychonoff theorem
Proof of Theorem 2.1. Fix A and W , and R 1 , . . . , R K > 0, K ∈ N. In addition, for simplification, we shall omit R 1 , . . . , R K in the notation for Step (I) In the first step we will approximate the operator V A,W . We shall discretize time, as we would like apply the classical Schauder-Tychonoff theorem in a compact subset which is given by a tight collection of laws on a finite time grid. Let us fix a sequence {ε ι : ι ∈ N} such that ε ι → 0.
First, let us introduce a dyadic time grid π n = {t 0 = 0 < t 1 < t 2 < · · · < t 2 n = T } by t k = T k 2 n , k = 0, 1, 2, . . . , 2 n . The stochastic process will be approximated by an averaging operator over the dyadic time interval. To this end, let us define a step-function . . , 2 n − 1 and T k 2 n ≤ s < T k+1 2 n , i.e., ϕ n (s) = T 2 −n ⌊2 n s⌋, s ≥ 0, where ⌊t⌋ is the largest integer that is less or equal t ∈ R. Let {w n : n ∈ N} ⊂ L m (Ω, F 0 , P; E) be a sequence, such that w n → w 0 in L m (Ω, F 0 , P; E) and For a function ξ ∈ M m A (X), we define (4.1) Note, that Proj n (ξ) is a progressively measurable, P-a.s. piecewise constant, U -valued stochastic process.
Remark 4.1. Observe that the projection operator satisfies (i) Proj n is a linear bounded contraction operator from X into X; (ii) If B is a bounded subset of X, then for all ε > 0 there exists a n 0 ∈ N such that Thus, due to Remark 4.1 and the uniform continuity of V A,W on the bounded set X (A), we know that for any ε > 0, there exists some n 0 ∈ N such that for every ξ uniformly in V A,W (X (A)) such that for any r ∈ (1, m], Let {n ι : ι ∈ N} be a sequence such that Finally, let us define the operator Step (II) Denote U := D([0, T ]; U ), the Skorokhod space of càdlàg paths in U endowed with the Skorokhod J 1 -topolgy, see [39,Appendix A2]. Given the probability space A = (Ω, F, F, P), for any ι ∈ N this operator V ι A,W induces an operator V ι on the set of Borel probability measures on X ∩ U, denoted by M 1 (X ∩ U). The construction of the operator V ι is done in the this step.
Define the subset of probability measures X given by Now, let µ ∈ X . Then, by the Skorokhod lemma [39,Theorem 4.30], we know that there exists a probability space A 0 = (Ω 0 , F 0 , P 0 ) and a random variable ξ : Ω 0 → X ∩ U such that the law of ξ coincides with µ. In particular, the probability measure ν ξ : B(X∩U) → [0, 1] induced by ξ and given by coincides with the probability measure µ.
Due to the definition of U, we know that ξ is a progressively measurable stochastic process, in particular, ξ : , P 0 ). Next, we have to construct the Wiener process and extend the probability space. Now, let A 1 = (Ω 1 , F 1 , (F t ) t∈[0,T ] , P 1 ) be a probability space where a cylindrical Wiener process W on H is defined, and let A µ the product probability space of A 0 and A 1 . In particular, we set Since µ ∈ X , we know ξ ∈ X (A µ ). Next, we have to verify if the family operators V ι Aµ,W : ι ∈ N is well-defined. However, this is follows from assumption (i), and since ξ ∈ X (A µ ). In fact, the µ-dependence of the stochastic basis can be removed by lifting to the space of probability measures (path laws). We aim to find a fixed point in the space of probability measures. Now, for A ∈ B(X ∩ U). Then, let us define the mapping V ι that maps the probability measure µ, in other words, the probability measure ν ξ : B(X ∩ U) → [0, 1] that is induced by ξ to the probability measure ν V ι Aµ,W (ξ) : B(X ∩ U) → [0, 1] given by . Note, since X∩U is a complete metric space, the space of probability measures over X∩U equipped with the Prokhorov metric 7 is complete.
The following points can be easily verified. (1) X is invariant under V ι . This follows directly from assumption (ii) and the properties of the projection Proj nι , see also Remark 4.1. Here Aα := {x ∈ X : d(x, A) < α}.
(2) Due to the fact that V ι Aµ,W restricted to X (A µ ) is uniformly continuous, V ι restricted to X is continuous on M 1 (X ∩ U) in the Prokhorov metric by [22,Theorem 11.7.1].
(3) Note that by assumption (v), V ι Aµ,W (ξ) ∈ U for ξ ∈ X (A µ ). We claim that V ι restricted to X is compact on M 1 (X ∩ U). In particular, it maps bounded sets into compact sets. In fact, we have to show that for all ι ∈ N and ε > 0 there exists a compact subset K ε ⊂ X ∩ U such that However, by assumption (iv) there exists a constant R > 0 with Due to the construction of the operator V ι , we have that the law is preserved. In particular, Next, by Chebyshev's inequality, we get that Since X ′ → X compactly, we have proved the tightness. (4) X is a convex subset of M 1 (X ∩ U). Let ν, µ ∈ X , we have to show that for any α ∈ (0, 1) we have αν + (1 − α)µ ∈ X . First, analyzing the expectation with respect to Ψ 1 , . . . , Ψ K , this follows by the linearity of the expectation value. Secondly, since ν, µ ∈ X we know that ν ({Θ < ∞}) = 1 and µ ({Θ < ∞}) = 1, Let α ∈ (0, 1). Then In particular, the mapping V ι restricted to X satisfies all assumptions of the classical Schauder-Tychonoff theorem, see [30, § 7, Theorem 1.13,. p. 148]: Lemma 4.2 (Schauder-Tychonoff). Let C be a nonempty convex subset of a locally convex linear topological space E, and let F : C → C be a compact map, i.e., F (C) is contained in a compact subset of C. Then F has a fixed point.
Step (III) Note, that the tightness argument in the previous step is independent of ι, thus the set {ν * ι : ι ∈ N} is tight, therefore there exists a subsequence {ι j : j ∈ N} and a Borel probability measure ν * such that ν * ι j → ν * , as j → ∞. In this step, we will construct from the family of probability measures {ν * ι j : j ∈ N} and ν * ∈ X , a filtered probability space A * , a Wiener process W * , a progressively measurable process w * , and a family of progressively measurable processes {w ι j : j ∈ N} that are P * -a.s. contained in X ∩ U over A * such that these objects have probability measures {ν * ι j : j ∈ N} and ν * ι j ∈ X ∩ U. By the Skorokhod lemma [39,Theorem 4.30], there exists a probability space A * 0 = (Ω * 0 , F * 0 , P * 0 ) and a sequence of X-valued random variables {w * ι j : j ∈ N} and w * ι j where the random variable w * ι j : Ω * 0 → X ∩ U has the law ν * ι j in X ∩ U. In addition, by tightness and the Skorokhod lemma, we have on X. Moreover, let us introduce the filtration G * 0 = (F * ,0 t ) t∈[0,T ] given by where N * 0 denotes the collection of zero sets of A * 0 . Next, similarly as above, let us construct the Wiener process. Let be a filtered probability space with a cylindrical Wiener process W * on H being adapted to the filtration ( T ], and P * = P * 0 ⊗ P * 1 . In addition, X (A * ) can be defined, and also the operators V ι A * ,W * and V A * ,W * Step (IV) In this step, we mimic an explicit Euler scheme, to construct a P * -a.s. piecewise constant and {G * t } t∈[0,T ] -progressively measurable process that is a fixed point for the operator V ι j A * ,W * . Since ν * ι j ∈ X , the process w * ι j ∈ X (A * ) and, hence, V However, we do not know if the process w * ι j satisfies In this step, we will construct here a fixed point for the operator V ι j A * ,W * . Let us define a new process by induction. To start with, let where t k ∈ π n are dyadic time points. Let us put w * Note, that by the definition of Proj ι j on [0, t ι j 1 ) the process on [0, t ι j 1 ) is defined by the initial data. In fact, we have for 0 ≤ s < t By equation (4.7) and equation (4.4) In particular, the process on [0, t ι j 1 ) is defined by the initial data and we have P * -a.s.
At time t ι j 1 , we have by equation (4.7) and equation (4.4), However, we havew * ι,1 (t Let us analyze what is happening at the next time interval [t ι j 1 , t ι j 2 ). Her the process is constant and equals P * -a.s. the value at t Note, also that P * -a.s. we have w * ι j ,1 (s) = w * ι j ,∞ (s) for t ι j 1 ≤ s < t ι j 2 , and, hence Let us analyze what happens in t ι j 2 . By equation (4.7), we have By equation of w * ι j ,2 , i.e. (4.5), we have Now, we can proceed by induction. Let us assume that in [0, t ι j k ) we have shown that Then, we have by equation (4.7) and equation (4.6) we have for t Step (V) Next, we verify a couple of statements with the goal to pass on to the limit. Here, we point out that the same construction as done for w * ι j ,∞ can be done on the initially given probability space A. The resulting process is denoted by w ι j ,∞ . Due to the construction and by the properties of the projection, it is easy to see that the laws are preserved. In particular, Law(w ι j ,∞ ) = Law(w * ι j ,∞ ). Claim 4.3. We claim that Proof of Claim 4.3: Clearly, since {w * ι j ,∞ } j∈N ⊂ X (A * ), and X (A * ) is bounded in X, we can conclude from the application of the Skorokhod lemma that E w ι j ,∞ r X = E * ∥w ι j ,∞ ∥ r X , for any r ∈ [1, m 0 ], so that we get by assumption (iv) that whereC > 0 is a constant such that ∥ · ∥ X ≤C∥ · ∥ X ′ which exists by the compact and dense embedding X ′ → X. Hence, we know that {∥w * ι j ,∞ ∥ r X } is uniformly integrable for any r ∈ (1, m 0 ] w.r.t. the probability measure P * . By (4.3), there exists w * ∈ X (A * ) with w * ι j ,∞ → w * P * -a.s., so we get by the Vitali convergence theorem that for any r ∈ (1, m 0 ). □ Step (VI) In this step we show that w * over A * together with the Wiener process W * is indeed a martingale solution to (2.2). We shall use an ε/3-argument and expand Now, we analyze the terms I, II, III, and IV separately. Due to the convergence, we know that for any r ∈ (1, m 0 ), ε > 0, there exists j 0 ∈ N, Next, to tackle II, we know due to the well-posedness by the existence of fixed point in the step before, V ι j A * ,W * (w * ι j ,∞ ) = w * ι j ,∞ . To tackle III, due to the uniform continuity of the operator V ι j A * ,W * , we know that there exists a δ = δ(ε) > 0 and j 0 ∈ N, such that, for any r ∈ (1, m 0 ), whenever j ≥ j 0 , and r X tends to zero by uniform continuity, see (4.2). However, m ≤ m 0 by assumption (iv). Thus, IV tends to zero in L r (Ω * , F * , P * ; X) for any r ∈ (1, m).
and the proof is complete. By construction, we see that w * solves

Results on regularity and technical propositions
This section contains the remaining results which are used in the proof of the main result Theorem 3.6.

5.1.
Assumptions on the noise and consequences. Let us recall, we denoted by {ψ k : k ∈ N} the eigenfunctions of the Laplace operator −∆ in L 2 (O) and by {ν k : k ∈ N} the corresponding eigenvalues, where the enumeration is chosen in increasing order counting the multiplicity.
Let us characterize the asymptotic behavior of {ν k : k ∈ N} and {ψ k : k ∈ N} for an arbitrary domain O with C 2 -boundary. Here, we know by Weyl's law [76,77] that there exist constants c, C > 0 such that, For instance, the eigenfunctions for L 2 ([0, 1] 2 ) for the multi-index k = (k 1 , k 2 ) ∈ Z 2 are given by the tensor product the corresponding eigenvalues are given by ν k = π 2 |k| 2 , where |k| = k 1 + k 2 . The case d = 3 works analogously. In this special case the conditions on δ 1 and δ 2 in Hypothesis 3.5 can be relaxed to where γ(H, E) denotes the space of γ-radonifying operators and | · | γ(H,E) the corresponding norm, cf. [7,73]. In case E is a Hilbert space, the γ-norm coincides with the Hilbert-Schmidt norm | · | L HS (H,E) . See Appendix C for further details.
Since |ξψ k | L m ≤ |ξ| L m |ψ k | L ∞ and (5.2), we know that Remark 5.2. From Hypotheses 3.5, one can infer that there exists a constant C > 0 such that Here {f k } is an orthonormal basis in H −1 2 (O), compare with [2, Hypothesis 3, p. 42]. In addition, note that is of linear growth and Lipschitz continuous. In particular, there exists constants C 1 , L 1 > 0 such that • Secondly, σ j : L 2 (O) → L HS (H j , L 2 (O)) is of linear growth and Lipschitz continuous. In particular, there exists constants C 2 , L 2 > 0 such that • Similarly, straightforward computations and using the fact that |f k | L ∞ ≤ ν d−1 2 k , see (5.2) and [2, p. 46], we get that where γ(H 2 , L m ) denotes the space of γ-radonifying operators, cf. [7,73]. (3.12). In this subsection, we are analyzing equation First, we will show that a unique solution u κ to the system (3.12) exists and is nonnegative. Second, we will show by variational methods that this solution satisfies some integrability properties, given u 0 ∈ L p+1 (O), p ≥ 1. Let us start with proving an inequality that we are going to use in the sequel. Lemma 5.3. Assume that the Hypotheses of Theorem 3.6 hold. For any ε 1 , ε 2 > 0 and any (ξ, η) ∈ X A (R 0 , R 1 , R 2 ) ⊂ M 2,m 0 A (Y, Z), u, w ∈ L γ+1 (O), κ ∈ N, there exists a constant C = C(κ, χ, ε 1 , ε 2 ) > 0 such that
We set where only ξ, and h(η, ξ, ·) depend on t and ω. and show its continuity. Note, that we have where only ξ, and h(η, ξ, ·) depend on t and ω. For the first integral in the above identity, we prove continuity with the same arguments as in [ Hence, the assumption (H1) of [48, Theorem 5.1.3] is satisfied.
, ω ∈ Ω. Take (A.1) into account and apply Lemma 5.3 to get that  Proof. For the nonnegativity of the solution to (3.12), we refer to the proof of positivity of the stochastic porous medium equation, see Section 2.6 in [2] and see also [1].
Mimicking the proof of nonnegativity in [2] and applying a comparison principle [42] the nonnegativity of (3.12) can be shown. □ In the next proposition we are using variational methods to verify uniform bounds of u κ , κ ∈ N.
Proposition 5.6. Fix p ≥ 1. Then, there exists a constant C 0 (p, T ) > 0 such that for every u 0 ∈ L p+1 (Ω, F 0 , P; L p+1 (O)), and for every κ ∈ N and for every (ξ, η) ∈ X A (R 0 , R 1 , R 2 ) such that u κ solves (3.12), the following estimate Proof. By Itô's formula for the functional u → |u| p+1 L p+1 , we get that u κ satisfies for t ∈ [0, T ], where t → M (t) is a local martingale. Integrating by parts, taking expectation (we may as well apply the Burkholder-Davis-Gundy inequality, see Section C) and rearranging yields, and hence by Remark 5.2 and Gronwall's lemma,

□
We may remark that the above result now permits an application of Proposition A.4. Note that the inequality becomes particularly useful, when u κ ≥ 0 and η ≥ 0. (3.13). Given the couple (η, ξ) ∈ M 2,m 0 A (Y, X), we consider the solution v κ to the equation (3.13). First, we will In the next proposition we investigate existence and uniqueness and the regularity of the process v κ . The constants R 0 , R 1 and R 2 are given as in (3.26), (3.23), and (3.25), respectively. In addition, for all κ ∈ N there exists a constant C = C(T, κ) > 0 such that

Properties of equation
Proof. Let us consider the following equation with a locally integrable and progressively measurable t → F (t) where the Laplace operator ∆ is equipped with periodic boundary conditions on a box or Neumann boundary conditions and initial condition w(0) = w 0 . A solution to (5.9) for F ≡ 0 is given by standard methods (see e.g. [16,Chapter 6] or [48,Theorem 4.2.4]) such that for any q ≥ 1, The solution to (5.9) is given by The Minkowski inequality, the smoothing property of the heat semigroup and the Sobolev embedding we obtain by the Hölder inequality for δ/2 < 1 − 1 µ and 2µ Observe, since ρ < 2 − 4 m − d 2 , such a δ and µ exists. Taking expectation, we know due to the assumptions on η, that the term can be bounded. It remains to calculate the norm in L 2 (0, T ; H ρ+1 2 (O)). By standard calculations (i.e. applying the smoothing property and the Young inequality for convolution) we get for 3 2 ≥ 1 µ + 1 κ and δκ/2 < 1 · 0 e rv∆(·−s) F (s) ds .
and similarly to before Proof. The heat semigroup, which is generated by the Laplace, maps nonnegative functions to nonnegative functions. In this way we refer to the proof of nonnegativity by Tessitore and Zabczyk [68]. The perturbation can be incorporated by comparison results, see [42]. □ Proposition 5.9. Assume that the Hypotheses of Theorem 3.6 hold. In addition let us assume that we have for the couple (η, Proof. We start to show (i). First, we get by the analyticity of the semigroup for δ 0 , δ 1 ≥ 0 Since, by the hypotheses, 2m 0 δ 0 < 1, the first term, i.e. I is bounded. In particular, we have Let us continue with the second term. The smoothing property of the semigroup gives for any δ 1 ≥ 0 Using the Sobolev embedding Supposing l δ 1 +r 0 2 < 1 the Young inequality for convolutions gives for and therefore, Which can be bounded by Hölder inequality as follows, Next, let us investigate III. We treat the stochastic term by applying [8, Corollary 3.5 (ii)], from which it follows forσ + r < 1 and β > 0 Due to the Sobolev embedding and interpolation, we know that whenever then there exists some θ ∈ (0, 1) such that Thus, ifρ satisfies (5.16), we get that Collecting everything together, choosing ε > 0 sufficiently small and subtracting εE∥v κ ∥ m 0 L m 0 (0,T ;L m ) on both sides, (i) follows. The rest of the proof is devoted to item (ii). Note, that for Substituting it in the definition of W α m 0 ([0, T ]; H −σ m (O)), see Appendix B, we can write First, let us note that by part (i) we know that there exists a number r > 0 and a constantC(T ) > 0 such that  In particular, since 2(r + δ) > α, the right hand side is bounded by estimate (5.17). Now, we consider the second term JJ. Jensen's inequality gives for any q > 1, q ′ := q q−1 , Hence, Taking into account that α < 1 m 0 ′ , integration and the Sobolev embedding gives It remains to give an estimate to the second term. We can show the assertion by the same computations as in the part of the proof for (i), i.e. starting at (5.15). Next, let us investigate JJJ. Here, again by [8, Corollary 3.5 (ii)], we get for some In this way we get E|JJJ| ≤2 Applying the Hölder inequality, we show that Collecting everything together gives (ii). □ Technical Proposition 5.10. Let us assume that (η, ξ) ∈ X A (R 0 , R 1 , R 2 ) ⊂ M 2,m 0 A (Y, Z). Then, for any α ∈ [1, m 0 2 ), q * ∈ (1, m 0 ], and κ ∈ N there exists some real numbers δ 1 , δ 2 ∈ (0, 1] and constants C 1 (κ, R 0 ), C 2 (κ, R 0 ) > 0 such that we have Proof. Firstly, observe that due to the cutoff function, at integrating with respect to the time we have to take into account in which interval s belongs to. Let τ u j := inf s>0 {h(η j , ξ j , s) ≥ κ} and τ o j := inf s>0 {h(η j , ξ j , s) ≥ 2κ}, j = 1, 2. Distinguishing the different cases, and noting that Lip(ϕ κ ) ≤ 2κ −1 , we obtain where τ u := min(τ u 1 , τ u 2 ) and τ o := max(τ o 1 , τ o 2 ). Therefore, we may integrate over [0, T ]. Secondly, observe that for any n ∈ N and any a, b ≥ 0, we have where the last inequality follows from an application of Young's inequality. The difference can be split in the following way for s ∈ [0, τ o ∧ T ] Therefore, we can write Taking expectation gives Similarly we get by the Hölder inequality Taking expectation we get Finally, take into account that , the assertion is shown. □ The next proposition gives the continuity property of the operator V κ,A .
Then, under the assumptions of Theorem 3.6 there exists a constant C = C(R 0 , κ) > 0 and real numbers δ 1 , δ 2 > 0 such that Proof. First, we get by the analyticity of the semigroup, Note, the assumptions on m 0 and m in the Hypothesis 3.4 gives the existence of l and µ. Now, by the technical Proposition 5.10 we can infer that there exists a constant C = C(R 0 , κ) > 0 and δ 1 , δ 2 > 0 such that It remains to tackle the second term II. This can be done by standard arguments using the Burkholder-Davis-Gundy inequality, see Section C. □ Next, we shall tackle the continuity of u κ with respect to η and ξ.
with u (j) κ (0) = u 0 , j = 1, 2. Then, under the assumptions of Theorem 3.6 there exist constants C = C(κ, R 1 , R 2 ) > 0 and c, δ 1 > 0 such that Proof. Applying the Itô formula and integration by parts gives where ∇ −1 := −(−∆) 1/2 . To deal with the second term and third term, i.e. II and III, we apply first the Burkholder Gundy inequality, resp. calculate the trace. Here, we apply Remark 5.2. Next, we obtain by the Young inequality and Lemma A.1 for Before tackling the first term I, let us introduce the following definition. Let us set and Ω c k := Ω \ Ω k . Now, we can decompose the third term into the following summands κ (s, x) dx ds =: I 1 (t) + I 2 (t) + I 3 (t) + I 4 (t) + I 5 (t) + I 6 (t).
Let us start with I 1 . Let us put for r, r ′ ,m,m ′ , p, p ′ , and s * = 2 γ+1 , Using duality withm ≥ γ + 1, 2 ≤ r < γ + 1, 1 m + 1 m ′ ≤ 1 and 1 r + 1 r ′ ≤ 1, we know for the integrand of I 1 By integration in time, and applying, firstly, the Hölder and, secondly, Young inequality we know for any ε 1 > 0 there exists a C(ε 1 ) > 0 such that Let us analyze the first term on the right hand side. Take some θ ∈ (0, 2/m) and let Then, we know by interpolation that

Hs r ′′
This gives The Hölder inequality gives .
By the Young inequality, we know for any ε 2 > 0 there exists some C(ε 2 ) > 0 such that .
Observe, for small θ, we know r ′′ ≥ r,s ≤ (s * − 1), andm ′′ := (1 − θ)m 2 2−θm ≥m. Thus we know, since In particular, δ = 3+2γ 2+6γ+4γ 2 . Now, to estimate the term by H ρ we need that < 0. In this way we get Hρ . It follows by taking into account the cut-off function In this way, we get for I 1 Next, we have to tackle I 2 (t). Again take r, r ′ ,m,m ′ , p, and p ′ defined as in (5.19). Again, using duality we know for the integrand of I 2 By the Young inequality we know for any ε 1 > 0 there exists a constant C(ε 1 ) > 0 such that Next, we follow the calculation as done before. First, we use again [58, p. 171, Theorem 1], where we put s 1 = −s * * < 0 and s 2 = s * * + ε, s * * = s * /2, ε > 0 very small. In this way we get Next, applying the Hölder inequality in time, Again, by the choice of r, r ′ ,m,m ′ , p, and p ′ , we know that even for a r ′′ > r ′ we have Again, similarly we know Now, taking into account the cut off function we know Since the operator is invariant, the second expectation value is smaller than R 1 . Next, let us treat I 3 .
By similar calculation as before, we get I 4 , I 5 , and I 6 can be estimated by similar steps. Now, we can collect all terms and get Term I cancels with the corresponding term on the left hand side. We may apply Gronwall's lemma in order to deal with the term III. □ Proposition 5.13. For any initial condition (u 0 , v 0 ) satisfying Hypothesis 3.4, (i) there exists r = r(T, γ) > 0 such that for any (η, ξ) ∈ X A (R 0 , R 1 , R 2 ), we have Proof. Let us prove (i) first. We know from Proposition 5.6 with p = 1, that for any (η, ξ) ∈ X A (R 0 , R 1 , R 2 ) we have On the other side, we know by Proposition A.4 that we have for any θ < 2 p+γ , where the LHS is bounded for p = 1, noting that Proposition 5.6 with p = γ gives Proof. Follows from an application of Fatou's lemma and nonnegativity of u κ and v κ in the proof of Proposition 5.6. □ Proposition 5.15. Let ρ < 1 − d 2 and assume that the Hypotheses of Theorem 3.6 hold. There exists a generic constant C(T ) > 0, a number l ≥ 2 |ρ| and δ 1 , δ 2 ∈ (0, 1) with δ 1 + δ 2 = 1 such that for all κ ∈ N we have Before proving Proposition 5.15, we consider the following lemma, which will be essential.
Here, we need α/p ′ < 1 which gives 2 ≤ p < α α−1 . Also, if Hρ . Taking the expectation and applying the Hölder inequality where we have to take into account that α/p ′ < 1 we get the assertion. □ Proof of Proposition 5.15: Let us consider the following equation for a locally integrable and progressively measurable t → F (t), Following the proof of Proposition 5.7 verbatim, we find that In particular, for α as in the Technical Lemma 5.16, we have 2 − 2 α > d 2 + ρ and α < 2. This gives as condition for ρ, ρ < 1 − d 2 . In addition, setting p = 2 we need −ρ < 2(2−α) α , which is not a restriction. However, due the hypotheses we find some α < 2 such that the first inequality is satisfied. Note, that due to the condition in Hypothesis 3.4, there exists some µ = α ≥ 1 such that ρ satisfies the assumption above and those of the Technical Lemma 5.16. Setting F = ϕ κ (h(η, ξ, ·))ηξ 2 , we obtain by applying Technical Lemma 5.16 that there exists some constant C(T ) > 0 and δ 1 , δ 2 ∈ (0, 1), δ 1 + δ 2 = 1 such that It remains to calculate the norm in L 2 (0, T ; H ρ+1 2 (O)). By standard calculations (i.e. applying the smoothing property and the Young inequality for convolution), we get for ) ≤ ∥F ∥ L µ (0,T ;L 1 ) , for and similarly to before we know by hypothesis 3.4, that there exists some µ = α such that δ − (ρ + 1) > d 2 , 3 2 ≥ 1 µ + 1 κ , δκ/2 < 1, and α(p − 1)/(p − α) ≤ m 0 . Therefore, we get by the Technical Lemma 5.16  i (t) − χu i (t)v 2 i (t) dt + σ 1 u i (t)dW 1 (t) t > 0, (6.1) dv i (t) = r v ∆v i (t) + u i (t)v 2 i (t) dt + σ 2 v i (t)dW 2 (t) t > 0. (6.2) In the first step we will introduce a family of stopping times {τ N : N ∈ N}, and show that on the time interval [0, τ N ] the solutions u 1 and u 2 , respective, v 1 and v 2 , are indistinguishable. Here, in the second step, we will show that P (τ N < T ) → 0 for N → ∞. From this follows that u 1 and u 2 are indistinguishable on the time interval [0, T ].