Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

We prove the existence of nonnegative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Gr\"un, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.


Introduction
In this work, we consider the stochastic thin-film equation where u = u(t, x) denotes the height of a thin viscous film depending on the independent variables time t ∈ [0, T ], where T ∈ (0, ∞) is fixed, and lateral position x ∈ T, where T is the one-dimensional torus of length L := |T|, and Q T := [0, T ] × T. Equation ( . ) describes the spreading of viscous thin films driven by capillary forces (acting at the liquid-air interface) and thermal noise and decelerated by friction (in the bulk or at the liquid-solid interface). The function M : R → [0, ∞) is called mobility and the following results apply to the choice M (r) = |r| n for r ∈ R, where n ∈ 8 3 , 4 . In particular, this covers the physically relevant case of a cubic mobility, that is, n = 3, modelling no slip at the liquid-solid interface in the underlying stochastic Navier-Stokes equations of which ( . ) is an approximation. The symbol W denotes a Wiener process in the Hilbert space H 2 (T).
Since its introduction over years ago, by Davidovitch, Moro, and Stone in [DMS ], and by the fourth author, Mecke, and Rauscher in [GMR ], the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise W in ( . ). The solution of this problem is the main result of this work.
We refer to [DG , ODB , BEI + ] for details on the physical derivation by means of a lubrication approximation and on the relevance of ( . ) in the deterministic case, where W = 0 in [0, T ] × T. Stochastic versions of the thin-film equation have been proposed independently in [DMS ] and [GMR ]. The former paper is concerned with the question of how thermal fluctuations enhance the spreading of purely surface-tension driven flow. On the contrary, the paper [GMR ] considers the effect of noise on the stability of liquid films and time-scales of the de-wetting process. Therefore, the energy considered in [GMR ] differs from that one of [DMS ] by an additional effective interface potential -giving rise to a so called conjoiningdisjoining pressure in the equation. We emphasize that the structure of the noise term in ( . ) is common to [GMR ] and [DMS ]. We further refer to [DOGKP ] for a more recent derivation of the model including the discussion of detailed-balance conditions.
A first existence result of martingale solutions to stochastic thin-film equations has been obtained in [FG ] by Fischer and the fourth author of this paper, in the setting of quadratic mobility M (r) = r 2 , additional conjoining-disjoining pressure, and Itô noise. We also mention the paper [Cor ] by Cornalba who introduced additional nonlocal source terms and in this way obtained results for more general mobilities. In [GG ], the second and the third author of this paper have studied ( . ) with Stratonovich noise and quadratic mobility M (r) = r 2 without conjoining-disjoining pressure. It turns out that non-negative martingale solutions exist that allow for touch down of solutions with complete-wetting boundary conditions. The case of quadratic mobility is special and simpler since in this case the stochastic part in ( . ) becomes linear. This I allows to separately treat the deterministic and stochastic parts in ( . ), a fact crucial to the approach in [GG ], and which fails in the case of non-quadratic mobility.
In this paper, we study the existence of weak (or martingale) solutions to ( . ) in the situation in which the gradient-noise term ∂ x √ M (u) • dW is nonlinear in the film height u, in particular covering the situation M (r) = |r| 3 . This includes precisely the situation studied in [DMS ] in the complete-wetting regime.
The analysis of the present work is based on a combination of estimates of the surface (excess) energy 1 2 T (∂ x u) 2 dx = 1 2 ∂ x u 2 L 2 (T) and the (mathematical) entropy T G 0 (u) dx, where G 0 (r) = r 2−n (2−n)(1−n) for r > 0, ∞ for r ≤ 0. ( . ) The main difficulty comes from the fact that -in contrast to the case of quadratic mobility and Stratonovich noise -the energy estimate cannot be closed on its own. This is caused by the nonlinear, stochastic conservation law structure of the noise in ( . ). Indeed, this nonlinear structure may lead to the occurrence of shocks and, hence, to the blow up of the energy 1 2 ∂ x u 2 L 2 (T) . In the light of this, the task becomes to understand if the thin-film operator, that is, the deterministic part in ( . ), has a sufficiently strong regularity improving effect to compensate the possible energy blow up caused by the stochastic perturbation. Since the thin-film operator degenerates when u ≈ 0, this requires a control on the smallness of u. Such a control is obtained by the entropy estimate, which explains its importance in the case of non-quadratic mobility. Indeed, in the present work we prove that a blow up of the energy can be ruled out by means of a combination of energy and entropy estimates. Once this importance of the entropy estimate for the construction of weak solutions to ( . ) is understood, the next task is to find approximations to ( . ) which allow for uniform (energy) estimates. In light of the previous discussion, these approximations are chosen in a careful way, compatible with both energy and entropy estimates.
We next give a brief account on the literature for the deterministic thin-film equation: A theory of existence of weak solutions for the deterministic thin-film equation has been developed in [BF , BBDP , BP ] and [Ott , BGK , Mel ] for zero and nonzero contact angles at the intersection of the liquid-gas and liquid-solid interfaces, respectively, while the higherdimensional version of ( . ) with W = 0 in [0, T ] × T and zero contact angles has been the subject of [DGG , Grü ]. For these solutions, a number of quantitative results has been obtained -including optimal estimates on spreading rates of free boundaries, i.e. the triple lines separating liquid, gas, and solid, see [HS , BDPGG , Grü , Fis ], optimal conditions on the occurrence of waiting time phenomena [DGG ], as well as scaling laws for the size of waiting times [GG , Fis ]. We also refer to [Grü ] for an existence result based on numerical analysis.
A corresponding theory of classical solutions, giving the existence and uniqueness for initial data close to generic solutions or short times, has been developed in [GKO , GK , GGKO , Gna , Gna , GIM ] for zero contact angles and in [Knü , KM , KM , Knü , Ess ] for nonzero contact angles in one space dimension, while the higher-dimensional version has been the subject of [Joh , Sei , GP ] and [Deg ] for zero and nonzero contact angles, respectively.
The paper is structured as follows: In § , we introduce the necessary mathematical framework and state our main result on existence of martingale solutions. In § we introduce a suitable approximation of ( . ) using a Galerkin scheme, a regularization of the mobility M controlled by a small parameter ε, and a cut-off in u L ∞ (T) . The Galerkin scheme only makes use of the energy inequality, which is valid also in the infinite-dimensional setting but ceases to hold as ε ց 0. In § we then derive an energy-entropy estimate which is uniform in ε and the cut-off in u L ∞ (T)

S M R
(the latter is removed at the end of this section). Finally, in § the limit ε ց 0 is carried out and the existence of martingale solutions to the original problem ( . ) is obtained.

Setting and Main Result . Notation
For a set X and A ⊆ X we write ½ A : X → {0, 1} for the indicator function of A, that is, For a measurable set D ⊆ R d , where d ∈ N, we write |D| for its d-dimensional Lebesgue measure. We write T := R/(LZ) for the one-dimensional torus of length L > 0. For any T ∈ [0, ∞) we write Q T := [0, T ] × T for the corresponding parabolic cylinder.

S M R
For s ∈ (0, 1) and u : U → X measurable, we define For k ∈ N 0 we define the periodic Sobolev space H k (T) as the closure of all smooth v : for the space H s (T) endowed with the weak topology.

. Setting
Suppose we are given a stochastic basis (Ω, F, F, P), that is, the triple (Ω, F, P) is a complete probability space and F = (F t ) t∈[0,T ] is a filtration satisfying the usual conditions. Further suppose that independent real-valued standard F-Wiener processes (β k ) k∈Z are given. For what follows, we write and n ≥ 1 is a fixed real constant called mobility exponent. Further assume that σ := (σ k ) k∈N is an orthogonal family of eigenfunctions for the negative one-dimensional Laplacian −∆ = −∂ 2 x on T (i.e., periodic boundary conditions are employed). Specifically, we introduce the orthonormal basis (e k ) ∞ k=−∞ of L 2 (T) with so that in particular We then write σ k =: ν k e k with ν k ∈ R ( . d) and assume k∈Z λ 2 k ν 2 k < ∞. ( . e) Notice that because of ( . b), this implies that The stochastic partial differential equation (SPDE) ( . ) thus attains the form It is more convenient for the subsequent analysis to rewrite this equation using Itô calculus, leading to a stochastic correction of the drift (in the physics literature sometimes referred to as the spurious drift), that is, .

Main Result and Discussion
We have the following notion of weak (or martingale) solutions to ( . ): such that (Ω,F ,F,P) is a filtered probability space satisfying the usual conditions,ũ (0) isF 0measurable and has the same distribution as u (0) , (β k ) k∈Z are independent real-valued standard F-Wiener processes, andũ is anF-adapted continuous H 1 w (T)-valued process, such that The main result of this paper reads as follows: where C < ∞ is a constant depending only on p, q, σ = (σ k ) k∈Z , n, L, and T . Moreover, u ∈ L p ′ (Ω,F,P; C γ 4 ,γ (Q T )), for all γ ∈ 0, 1 2 and p ′ ∈ 1, 2p n+2 .
The proof of Theorem . is given in Section . below. Theorem . is a global existence result for weak solutions to the stochastic thin-film equation ( . ) for a range of mobility exponents, including the cubic one n = 3, corresponding to a no-slip condition at the substrate of the underlying stochastic Navier-Stokes equations (see [DMS ] for details on the modelling and a non-rigorous derivation). Therefore, Theorem . in particular applies to the physically relevant situation considered in [DMS ]. We expect that the limitations n ≥ 8 3 and n < 4 are due to technical reasons and that these restrictions can be potentially removed in future work by making use of so-called α-entropies as first introduced in [BBDP ]. Similarly, upgrading Theorem . to cover higher dimensions, as done in [DGG , Grü ], would be an interesting direction for future research. Notably, our solutions are nonnegative as in [GG ] but since G 0 (ũ(t)) L 1 (T) is dt ⊗ dP-almost everywhere finite, by ( . ) it holds |{ũ(t) = 0}| = 0 for all t ∈ [0, T ], dP-almost everywhere. Since the arguments in [GG ] are purely energetic, the support of the initial data in [GG ] is not necessarily T and in general this is not the case for the corresponding solution of the SPDE, either. We expect that it is possible to overcome this constraint also in the situation of this paper by using a renormalization technique, which will be left as an endeavour for future research, too.

Galerkin Approximation
In this section, we use the definitions and assumptions of § . .

. Setup
We write V N = span{e −N , . .., e N }, where the (e j ) j∈Z are defined as in ( . a), N ∈ N, and let Π N : L 2 (T) → V N be the orthogonal projection given by It is immediate from ( . b Furthermore, we obtain for any v ∈ L 2 (T) through integration by parts and with our specific choice of eigenfunctions, Let g : [0, ∞) → [0, 1] be a smooth function such that g = 1 on [0, 1] and g = 0 on [2, ∞).
where n > 0 is constant. Notably, for ε = 0 the definition ( . ) is consistent with the corresponding expression in ( . ), but we will assume ε > 0 and thus that F ε is smooth with F ε (r) ≥ ε n 2 for all r ∈ R until § . We consider the Galerkin scheme, i.e., the finite-dimensional stochastic differential equation (SDE) The approximation in ( . ) is three-fold. While applying the projection Π N yields a finitedimensional SDE, additionally the mobility F 2 0 is regularized with F 2 ε , so that the limiting equation as N → ∞ is non-degenerate if ε > 0. For technical reasons in what follows, we also cut off the noise with the pre-factor g R u ε,R,N L ∞ (T) .
Notice that ( . ) is equivalent to the system on R 2N +1 where, with the short-hand notation v y (x) = N j=−N y j e j (x) for y ∈ R 2N +1 , Let us consider on R 2N +1 the inner product and denote by · λ the corresponding norm. By ( . b), it is easy to see that for all y ∈ R 2N +1 we have In addition, because of the truncation in R and the finite dimensionality, it is easy to see that there exists a constant C = C(R, N ) such that for all y ∈ R 2N +1 This shows that the system ( . ) is coercive, which combined with the local Lipschitz continuity of the coefficients implies that for any F 0 -measurable random variable in R 2N +1 , there exists a unique solution of ( . ) starting from y 0 . In particular, ( . ) has a unique solution starting from u (0) N := Π N u (0) . Finally, notice that with ( . ) it follows that ( . ) is still in divergence form so that in particular A(u ε,R,N (t)) = A(u ε,R,N (0)) for any t ∈ [0, T ].

.
Energy Estimate for the Galerkin Scheme Lemma . . Suppose p ∈ [2, ∞), u (0) ∈ L p Ω, F 0 , P; H 1 (T) , and n > 0. Let u ε,R,N be the unique solution to ( . ) with initial data u (0) N . Then u ε,R,N satisfies where C < ∞ is a constant depending only on ε, R, p, σ = (σ k ) k∈Z , n, and T (but not on N ). Proof. For convenience, we drop the dependence on ε, R, and N in the notation and simply write u and γ u (t) := g R u(t) L ∞ (T) . Applying Itô's formula to ( . ), we have, dP-almost surely, , we obtain with the help of ( . ) the simplification Integration by parts gives for the terms to the right of the inequality This and our control of u L ∞ (T) via the cut-off function γ u imply together with ( . ) that Consequently, by Young's inequality we have Let us set By replacing t with t ∧ τ m in the above inequality, raising to the power p 2 , taking expectations, and using Grönwall's lemma, we conclude that The Burkholder-Davis-Gundy inequality and the Cauchy-Schwarz inequality imply which shows that the last term at the right hand side of ( . ) can be dropped. The claim then follows by letting m → ∞ and using Fatou's lemma. .

Passage to the Limit in the Galerkin Scheme
Let us consider the equation such that (Ω,F ,F,P) is a filtered probability space satisfying the usual conditions,û (0) isF 0measurable and has the same distribution as u (0) , (β k ) k∈Z are independent real-valued standard F-Wiener processes, andû ε,R is anF-adapted continuous H 1 (T)-valued process, such that Remark . . . Note that Definition . covers also the case that the cutoff by g R is not activejust by formally setting R = ∞.

. (Mass conservation) In the situation of Definition . by setting
Hence, by Poincaré's inequality there exists a constant C L < ∞, only depending on L, such that we have Proposition . . For n ∈ (0, 4], p ≥ n + 2, and u (0) ∈ L p Ω; F 0 , P; H 1 (T) , problem ( . ) admits a weak solution in the sense of Definition . .

G A
Proof. Let (Ω, F, F, P) be a filtered probability space carrying a sequence β k ∞ k=1 of independent F-Wiener processes and on this probability space let u ε,R,N be the unique -probabilistically-strong solution of ( . ). From now on, since ε and R are fixed, we drop them and we write u N instead of u ε,R,N in order to simplify the notation. By Lemma . we have that u N satisfies the bound where C < ∞ is independent of N . Let us introduce the notation γ w (t) (recall that we can interchange the projection operator and the derivative by virtue of ( . )). Let α ∈ 0, 1 2 such that α > 1 p . By Sobolev's embedding and Young's inequality, we have where we have used 2n − 2 ≤ n + 2. By [FG , Lemma . ] we get where (e k ) k∈Z is the standard orthonormal basis of ℓ 2 (Z). We now fix s ∈ 1 2 , 1 . By [Sim , § , Corollary ] we have that the embedding is compact. Combining this with ( . ) and ( . ), it follows that for each δ > 0 a com- and for each N ∈ N, as random variables in Z It follows thatθ We setû (0) :=û(0, ·). LetF = (F t ) t∈[0,T ] be the augmented filtration of It follows thatβ k , k ∈ Z, are mutually independent, standard, real-valuedF t -Wiener processes (see, e.g., [GG , Proposition . ] or [DG , Proof of Proposition . ] or [FG , Lemma . ]). We claim that the probability space (Ω,F,F, P) withF :=F T , together with the Wiener processes (β k ) k∈Z and the processû set up a weak solution of ( . a). Notice that Definition . (i) is satisfied because of ( . ), ( . ), ( . ), and Fatou's lemma. Hence, we only have to prove Definition . (ii) and the continuity ofû as a process with values in in H 1 (T). Let us set Fix an arbitrary l ∈ Z. We will show that for any ϕ ∈ H −1 (T), the processes are continuousF t -martingales. We first show that they are continuous G t -martingales. Let us further assume for now that ϕ ∈ N ∈N V N , and for i = 1, 2, 3 and v ∈ {u N ,û N }, let us also define the processes which combined with ( . ) giveŝ By ( . ) we have that dP-almost surely û N −û C(Q T ) → 0 as N → ∞, which in particular implies that dP-almost surely in L 2 (Q T ) Since in addition from ( . ) and ( . ) we have that dP-almost surely in L 2 (Q T ) one easily deduces that for each t ∈ [0, T ], dP-almost surely In addition, we have where in the last step we have used qn 2 = p, Sobolev's inequality, and ( . a) combined with conservation of mass. Similarly, for q := 2p n > 2, from which one deduces that for each i = 1, 2, 3 and t ∈ [0, T ], the M i N (û N , t) are uniformly integrable inω ∈Ω. Hence, we can pass to the limit in ( . ) to obtain In addition, using the continuity of M i (û, t) in ϕ, the uniform integrability inΩ, and the fact that N V N is dense in H −1 (T), it follows that ( . ) holds also for all ϕ ∈ H −1 (T). Hence, for all ϕ ∈ H −1 (T), i = 1, 2, 3, one can see that theM i (û, t) are continuous G t -martingales having finite q 2 -moments, where q := 2p n . In particular, by Doob's maximal inequality, they are uniformly integrable (in t ∈ [0, T ]), which combined with continuity (in t ∈ [0, T ]) implies that they are alsoF t -martingales. By [Hof , Proposition A. ] we obtain that dP-almost surely, for Choosing ϕ := (I − ∆)ψ in ( . ) for ψ ∈ C ∞ (T), we obtain that for dP ⊗ dt-almost all By [KR , Theorem . ] we have thatû is anF-adapted continuous L 2 (T)-valued process and therefore the above equality is satisfied dP-almost surely, for all t ∈ [0, T ]. Moreover, from the above and the fact thatû satisfies Definition . (i), it follows that for all ψ ∈ C ∞ (T), for almost all (ω, t) ∈Ω × (0, T ), we have is a predictable H −2 (T)-valued process such that with probability one v * ∈ L 2 ((0, T ); H −2 (T)), M (û, ·) is an L 2 (T)-valued martingale, and the duality between H 2 (T) and H −2 (T) is given by means of the inner product in L 2 (T). Hence, ∂ xû also satisfies the conditions of [KR , Theorem . ] with the choices V = H 2 (T) and H = L 2 (T). Consequently, ∂ xû is also continuous L 2 (T)-valued. This finishes the proof.

A-priori Estimates
In this section, we use the definitions and assumptions of § . .

. Entropy Estimate
For r ∈ R, let us set where F ε (r) was introduced in ( . ). We collect some properties of F ε , G ε , and H ε that we will need later on.
Lemma . . Let n > 2. Then there exists a constant C n < ∞, only depending on n, such that for all r ∈ R and all ε ∈ (0, 1) we have A-E Proof. Let us first look at the case r ≥ 0. We have where for the last inequality we have used ( . ). Hence, we only have to check the case r < 0. In this case we have since F ε is even. Therefore, where we have used ( . ) and the fact that G ε is decreasing. This finishes the proof.
Proof. For the convenience of the reader, we simply writeû instead ofû ε,R . By Itô's formula , so that after integration by parts we get dP-almost surely. Then we have for all δ > 0 k∈Z T where for the last inequality we have used Lemma . . Moreover, since ∂ xû has zero average, we get from Poincaré's inequality using conservation of mass (cf. Remark . ), Consequently, for any δ > 0 we have Using this in ( . ), choosing δ > 0 small, rearranging, and taking the p-th power gives is the martingale from ( . ). Notice that G ε (û(t)) is a continuous L 1 (T)-valued process and let us set Taking suprema up to τ m ∧ t ′ , for t ′ ∈ [0, T ], in the above inequality and expectation, we obtain by virtue of the Burkholder-Davis-Gundy inequalitŷ A-E Next, by integration by parts, we havê Using this and rearranging in ( . ), we have the desired inequality by virtue of Grönwall's inequality and Fatou's lemma. .

Uniform Energy Estimate
The following auxiliary result is convenient for deriving an energy estimate.

Further notice that
so that we may infer that ( . ) holds true.
Proof. For convenience of the reader, we writeû instead ofû ε,R . By Itô's formula (see, e.g., [Kry ]) we have dP-almost surely, where we write γû(t) := g R û(t) L ∞ (T) . The same reasoning as in the proof of Lemma . leads to dP-almost surely.
(∂ xû ) 4 -term We first focus on estimating the term k∈Z t 0 γ 2 u T σ 2 k (F ′′ ε ) 2 (û) (∂ xû ) 4 dx dt ′ and note that through integration by parts we have With help of the Cauchy-Schwarz and Hölder's inequality we have Next, since ∂ xû has zero average, we use that In the case n ∈ 8 3 , 3 , we deduce with help of Lemma . that dP-almost surely, where ϑ > 0 and we have applied conservation of mass (cf. Remark . ) and the second Poincaré inequality. Integration in time yields for any ϑ > 0, dP-almost surely. Applying Young's inequality and confining ϑ to the interval (0, 1) yields for any dP-almost surely.
This implies and Lemma . yields For n ∈ [3, 4), by employing mass conservation (Remark . ) and the Sobolev embedding, we get for any δ > 0 dP-almost surely, where we have applied Young's inequality in the last step.
For n ∈ 8 3 , 3 , with an analogous reasoning we obtain for any δ > 0 dP-almost surely, where we have applied Young's inequality again. Altogether, for n ∈ 8 3 , 4 , we obtain dP-almost surely.
Then, we estimate, using conservation of mass (Remark . ) and the Sobolev embedding, L 2 (T) + |A(û (0) )| n dP-almost surely, where we have also employed that ∂ xû has zero average. Integrating in time yields dP-almost surely, so that by Young's inequality it follows that for any δ > 0, dP-almost surely.
Closing the estimate Inserting all the previous estimates ( . ), ( . ), ( . ), and ( . ) in ( . ) and choosing δ sufficiently small and an appropriate ϑ, we arrive for n ∈ 8 3 , 4 at dP-almost surely, where q ≥ max 1 4−n , n−2 2n−5 and q > 1 with a constant C q,σ,n,,L,T,δ < ∞. Here, denotes the martingale in the last line of ( . ). Let us set as usual We now discard the second term on the left-hand side of ( . ), we take suprema in time up to T ∧ τ m , we raise to the power p 2 , and we take expectation to obtain with help of the Burkholder-Davis-Gundy inequalitŷ T ∧τm +C p,q,σ,n,L,T 1 +Ê sup Now we go back to ( . ), we discard the first term at the left-hand side and we conclude that T ∧τm + C p,q,σ,n,L,T 1 + |A(û (0) )| n +Ê sup dP-almost surely. From this it follows with Young's inequality that +C p,q,σ,n,L,T Ê sup which combined with ( . ) and ( . ) yields by virtue of Fatou's lemmâ which was stated in ( . ). .

Passage to the Limit to remove the Cut Off
In this section we consider the SPDE The definition of a weak solution {(Ω,F ,F,P), (β k ) k∈Z ,ǔ (0) ,ǔ} of equation ( . ) is covered by Definition . taking for g R the function g ∞ ≡ 1.
Proof. Letû ε,R be a weak solution of ( . a). By Lemmata . and . we havê L 2 (Q T ) ≤ C p,q,σ,n,L,T 1 + K(u (0) , p, q, ε) , and notice that the constant does not depend on R. From this estimate, the construction of a weak solution {(Ω,F ,F,P), (β k ) k∈Z ,ǔ (0) ,ǔ} is very similar to the construction in Proposition . (in fact, easier) and is left to the reader. Estimate ( . ) follows from the above estimate and Fatou's lemma.

The Degenerate Limit
In order to prove Theorem . , we first prove additional regularity in time in order to obtain dP-almost surely uniform convergence in the limit ε ց 0 using a version of Prokhorov's theorem (cf. [Jak , Theorem ]) and a compactness argument. Subsequently, we prove that ( . ) is recovered in this limit by employing the energy-entropy estimate, Proposition . . The proof is T D L concluded by showing that the weak formulation ( . ) is valid, which follows by applying [Hof , Proposition A. ] to characterize the martingale. For ε ∈ {n −1 } ∞ n=1 , we denote by {(Ω ε ,F ε ,F ε ,P ε ), (β k ε ) k∈Z ,ǔ (0) ε ,ǔ ε } the weak solution of ( . a) constructed in Proposition . . In order to drop the ε-dependence from the probability space we will be considering ((β k ε ) k∈Z ,ǔ (0) ε ,ǔ ε ) on a common probability space given by

. Compactness
The reasoning of this section uses techniques of [FG ] and of [GG , § ].
We then use that valid for any σ ∈ 0, 1 2 . For R 1 , we estimate using F ε (r) ( . ) ≤ (r 2 + ε 2 ) n 4 , Hence, The term R 2 2 is split as follows For R 21 , we estimate Finally, Hence, we geť Altogether, combining ( . ), ( . ), ( . ) and choosing σ = 1 4 , we get which gives the desired estimate ( . b). Note that we have used Proposition . to get an estimate in terms of the initial data.
By interpolation, we get the following result on Hölder regularity with respect to space and time (for details, see [FG , Lemma . , p. ]). Note that in Corollary . , we control the T D L moment of order p ′ of ǔ ε C for any p ′ ∈ [1, 2p) while in [FG ] only estimates for second moments have been provided. This is due to ( . a) as the analogous estimate in [FG ] has been formulated only for p ′ = 2.
In the next proposition we will consider the space of all functions u : Q T → R such that u ∈ C γ 4 ,γ (Q T ) for all γ ∈ (0, 1/2), that is, We endow C 1 8 −, 1 2 − (Q T ) with the topology generated by the metric Moreover, there exists a countable set D ⊂ C ∞ (Q T ) such that for all v ∈ C ∞ (Q T ) and all ε > 0, there existsṽ ∈ D such that ṽ − v C 1 (Q T ) ≤ ε. It follows that D is dense in C 1 8 −, 1 2 − (Q T ). In addition, it is complete, since C γn 4 ,γn (Q T ) is complete for each n. Therefore it is a Polish space.
Proof. It suffices to show the tightness of the laws µǔ ε , µJ ε , and µW ε corresponding to the familieš u ε ,J ε , andW ε . The proposition then follows by applying [Jak , Theorem ].
Tightness of the law µW follows because µW is a Radon measure in the Polish space C [0, T ]; H 2 (T) implying regularity from interior and thus tightness.
Tightness for µǔ ε as a family of measures on C 1 8 −, 1 2 − (Q T ) is a direct consequence of Corollary . , in particular estimate ( . ) (see also Remark . ) .
By Markov's inequality we have for any R ∈ (0, ∞), using conservation of mass (cf. Remark . ) and the Sobolev embedding theorem, where we have used Proposition . . Finally, ( . ) holds by virtue of ( . a) and Corollary . . This finishes the proof.
For what follows, we defineF = (F t ) t∈[0,T ] as the augmented filtration of (F ′ t ) t∈[0,T ] , wherẽ We further defineβ Then, we work in the filtered probability space Proof. The reasoning is quite standard and contained in detail for instance in [GG , Proposition . ] or [DG , Proof of Proposition . ] or [FG , Lemma . ].

Proposition . . Assume the conditions of
In particular, the following energy-dissipation estimate holdsẼ Proof. The proof follows the lines of the proof of [GG , Proposition . ], with the additional complication of taking care of the approximation F ε (r) of the square root of the mobility F (r) = |r| n 2 .
T D L .

Recovering the SPDE
In this section, we give the proof of the main result Theorem . : Proof of Theorem . . We first note that the results of § . can be applied due to the assumptions of Theorem . , with p in § . replaced by p n+2 from the statement of Theorem . . We will show that {(Ω,F,F,P), (β k ) k∈Z ,ũ(0),ũ} is a solution of ( . ). The fact thatũ(0) has the same distribution as u (0) follows from Proposition . and ( . ) therein. By Proposition . , u is anF-adapted continuous H 1 w (T)-valued process, so that in particularũ(0) isF 0 -measurable. The fact that theβ k are independent real-valued standardF-Wiener processes is the content of Lemma . . The Hölder regularity stated in ( . ) is a consequence of ( . ) of Proposition . . Moreover, (i) and (ii) from Definition . follow from ( . ) and ( . ). Hence, we only have to show (iii).
T D L Argument for ( . b) By Proposition . , we can identify so that the limit in ( . b) follows from ( . ),ũ ε →ũ in C(Q T ), andJ ε ⇀J in L 2 (Q T ), dP-almost surely.
In order to complete the proof, we only have to show that