The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential

We prove existence of martingale solutions for the stochastic Cahn-Hilliard equation with degenerate mobility and multiplicative Wiener noise. The potential is allowed to be of logarithmic or double-obstacle type. By extending to the stochastic framework a regularization procedure introduced by C. M. Elliott and H. Garcke in the deterministic setting, we show that a compatibility condition between the degeneracy of the mobility and the blow-up of the potential allows to confine some approximate solutions in the physically relevant domain. By using a suitable Lipschitz-continuity property of the noise, uniform energy and magnitude estimates are proved. The passage to the limit is then carried out by stochastic compactness arguments in a variational framework. Applications to stochastic phase-field modelling are also discussed.


Introduction
The Cahn-Hilliard equation was firstly proposed in [14] in order to describe the spinodal decomposition occurring in binary metallic alloys. Since then it has been increasingly employed in several areas such as, among many others, physics, engineering, and biology. In the recent years, the Cahn-Hilliard equation has become one of the most important models involved in phase-field theory. In such class of models, the evolution of a certain material exhibiting two different features is described by introducing a so-called state variable ϕ ∈ [−1, 1], representing the difference in volume fractions. The sets {ϕ = 1} and {ϕ = −1} correspond to the pure regions, while the interfacial region {−1 < ϕ < 1} where the two components coexist is supposed to have a positive thickness. For this reason, such models are usually referred to as diffuse interface models, and the time evolution of the state variable is often described by means of a Cahn-Hilliard-type equation. The field of applications of diffuse-interface modelling is enormous. In physics it is used in the context of evolution of separating materials, phase-transition phenomena, and dynamics of mixtures of fluids; in biology phase-field modelling is crucial in the description of evolution of interacting cells, tumour growths, and dynamics of interacting populations; in engineering it plays a central role in modelling of damage and deterioration in continuous media.
Given a smooth bounded domain O of R d , with d ≥ 2, and a fixed final time T > 0, the deterministic Cahn-Hilliard equation reads n · ∇ϕ = n · m(ϕ)∇µ = 0 in (0, T ) × ∂O , (1.3) The variable ϕ is referred to as state variable, or order parameter, while µ is the chemical potential.
Here, the symbol n denotes the outward unit vector on the boundary ∂O, the function m is known as mobility, while F : R → [0, +∞] is a double-well potential with two global minima. Typical examples of m and F are given below.
The chemical potential µ is directly related in equation (1.2) to the subdifferential of the so-called free-energy functional The double-well potential F may be thought as a singular convex function that has been perturbed by a concave quadratic function: the effect of the concave perturbation is then the creation of two global minima for F , each one representing the pure phases of the model. Minimising the F -term in freeenergy above describes then the tendency of each pure phase to concentrate, whereas the gradient term penalises high oscillations of the state variable. The idea behind the minimisation of the free-energy is then a calibration between these two phenomena.
Since Here, by contrast, the global minima corresponds exactly to the pure phases ±1, and this choice is then often employed in the modelling contexts where the pure phases have a privileged role compared to the interface: it is the cases, for example, of tumor growth dynamics in biology. In (1.6), the derivative F ′ ob has to be interpreted in the sense of convex analysis as the subdifferential ∂F ob , and the equation takes the form of a differential inclusion. In some cases, these double-well potentials are approximated by the polynomial one F pol (r) := 1 4 (r 2 − 1) 2 , r ∈ R , (1.7) which nonetheless does not ensure the relevant constraint ϕ ∈ [−1, 1]. While on the one hand the polynomial potential F pol is certainly much easier to handle from the mathematical point of view, on the other hand the logarithmic potential F log is surely the most relevant in terms of thermodynamical consistency. Indeed, due to the physical interpretation of diffuse-interface modelling, only the values of the variable ϕ in [−1, 1] are meaningful. For this reason, the possibility of dealing with the logarithmic potential F log is crucial.
The classical choice of the mobility m is a positive constant m con , independent of ϕ. Nevertheless, starting from the pioneering contribution [10] itself, several authors proposed the choice of a mobility depending explicitly on the order parameter. A thermodynamically relevant choice for m has been exhibited in the works [11,12,56]  Several variants of the Cahn-Hilliard equation have been studied in the last decades. Novick-Cohen proposed in [67] the viscous regularization (see also [41,42]), accounting also for viscous dynamics occurring in phase-transition evolution. Gurtin generalized the viscous correction in [55] possibly including nonlinear viscosity contributions in the equation. More recently, physicists have introduced the so-called dynamic boundary conditions in order to account also for possible interaction with the walls in a confined system (see for example [45,50,59]).
The mathematical literature on the deterministic Cahn-Hilliard equation is extremely developed: we refer to [65] and the references therein for a unifying treatment on the available literature. In particular, in the case of constant mobility existence, uniqueness, and regularity have been studied in [15,16,18,19,21,51] both with irregular potentials and dynamic boundary conditions, and in [7,8,66,75] with nonlinear viscosity terms. Significant attention has been devoted also to the asymptotic behaviour of solutions [20,24,52] and optimal control problems [17,22,23,57]. A mathematical analysis of the framework of nonconstant and possibly degenerate mobility has been investigated in [13,40]. In this direction we also refer to the contribution [29,62,84] regarding existence of solutions. Let us point out the work [54] dealing with the analysis of a Cahn-Hilliard equation with mobility depending on the chemical potential, [78] in relation to global attractors, and [61] for an approach based on gradient flows in Wasserstein spaces. A diffuse interface model with degenerate mobility has been studied also in [48]. Numerical simulations have been analyzed in [4,60].
Despite the fact that the deterministic model has been proven to be extremely effective in the description of phase-separation, there are certainly important downsides. One of the main drawbacks of the deterministic framework is the impossibility of describing the unpredictable disruptions occurring in the evolution at the microscopic scale. These may be due to several phenomena of different nature, such as uncertain movements at a microscopic level caused by configurational, electronic, or magnetic effects, which cannot be captured by the classical deterministic Cahn-Hilliard system. The most natural way to capture the randomness component which may affect phase-field evolutions is to introduce a Wienertype noise in the Cahn-Hilliard equation itself. This was first proposed in [25] employing Wiener noises in the well-known Cahn-Hilliard-Cook stochastic version of the model, which has been then validated as the only genuine description of phase-separation in the contributions [6,71]. Currently, this is widely studied both in the physics and applied mathematics literatures, for which we refer to [64,72] and the references therein. The stochastic version Cahn-Hilliard equation reads Here, W is a cylindrical Wiener process defined on certain stochastic basis, while G is a suitable stochastically integrable operator with respect to W . Precise assumptions on the data are given in Section 2 below.
From the mathematical point of view, the stochastic Cahn-Hilliard equation has been studied mainly in the case of polynomial potential and only with constant mobility. One the first contributions in this direction is [27], in which the authors show existence of solutions via a semigroup approach in the case of polynomial potentials. More recently, well-posedness has been investigated also in [26,39] again in the polynomial setting. A more general framework allowing for rapidly growing potentials (e.g. more than exponentially) has been analyzed in [74,77] from a variational approach. The genuine case of logarithmic potentials has only been covered in the works [30,31,53] by means of so-called reflection measures.
The mathematical literature on stochastic phase-field modelling has also been increasingly developed. Let us point out in this direction the works [1] dealing with unbounded noise, [43,44] for a study of a diffuse interface model with thermal fluctuations, and [5,69] dealing with the stochastic Allen-Cahn equation. Beside well-posedness, optimal control problems have also been studied in [76] in the case of the stochastic Cahn-Hilliard equation, and in [68] in the context of a stochastic phase-field model for tumour growth. Besides, let us point out the mathematical literature on stochastic two-phase flows has also been expanded in the last years, in the context of coupled stochastic systems of Cahn-Hilliard-Navier-Stokes and Allen-Cahn-Navier-Stokes type. In this direction, we refer the reader to the contributions [33,35,80,81] for existence of solutions, [3,36] about asymptotic long-time behaviour, [37] on large deviation limits, and [32,34] dealing with a nonlocal phase-field equation in the system instead.
One of the most critical problems of the stochastic model is that the presence of the random forcing term does not guarantee that the state variable ϕ remains in the physically relevant domain [−1, 1], even if the double-well potential is singular as in (1.5) or (1.6). This is due to the presence of the additional second-order term in the energy balance, which may cause blow-up of the energy in finite time. The consequences of this fact are sever, both on the mathematical side and especially from the modelling perspective. Indeed, from the point of view of thermodynamically consistency of the equation, ϕ represents a local concentration and is only meaningful if belonging to the physical interval [−1, 1]: the impossibility of proving that the solution of the stochastic equation satisfies this constraint inevitably represents a modelling downside. Besides, as we have pointed out above, the available literature on the stochastic Cahn-Hilliard equation only handles the case of constant mobility, which unfortunately is not the most suited for describing phase-separation.
The current state-of-the-art of the stochastic Cahn-Hilliard equation inevitably calls then for a deeper investigation in the direction of degeneracy of the mobility, including the physically relevant logarithmic potential, and showing that the physically meaningful constraint ϕ ∈ [−1, 1] is achieved. This paper is the first contribution in the mathematical literature that addresses these three points, and represents then an important step especially in terms of applicative validation of the stochastic Cahn-Hilliard-Cook model. Our results have important consequences to all fields, in particular physics and engineering, where phase-separation is usually studied under random forcing, as we provide the first mathematical validation of the stochastic model in its more relevant form, i.e. with degenerate mobility m pol and logarithmic potential F log .
In this paper we are interested in studying the stochastic Cahn-Hilliard equation (1.10)-(1.13) from a variational approach, including the cases of degenerate mobility m pol , the logarithmic double-well potential F log , and the double-obstacle potential F ob . As we have pointed out, these choices are indeed the most relevant in terms of the thermodynamical coherency of the model. From the mathematical perspective, our approach differs from [30,31,53] as we do not rely on reflection measures. Our techniques extend to stochastic framework the ideas of C. M. Elliott and H. Garcke in [40], and consist of a compatibility condition between the degeneracy of the mobility, the coefficient G, and the possible blow-up of the potential at the extremal points ±1.
Let us briefly explain the main difficulties arising in the case of degenerate mobility and logarithmic potential, and how we overcome these in the present work.
The first main issue appearing in the stochastic framework is the presence of a proliferation term in equation (1.10). Indeed, in the deterministic case the integration in space of equation (1.1) yields, together with the boundary conditions (1.3), the conservation of mass during the evolution. This is in turn crucial when dealing with irregular potentials such as F log or F ob , as it allows to control the spatial mean of the chemical potential. However, in the stochastic scenario the presence of the noise term in equation (1.10) determines a proliferation of the total mass of the system. Whereas this drawback can be overcome in the easier case of regular potentials as F pol , this results in the impossibility of obtaining satisfactory estimates on the chemical potential µ in the case of logarithmic and doubleobstacle potentials. The main reason is the following. On the one hand the derivatives of F pol can be controlled by F pol itself, i.e. |F ′′ pol |, |F ′ pol | ≤ c(1 + F ) for a certain c > 0, so that the usual energy estimates on F pol allow to bound also F ′ pol , hence µ as a byproduct. On the other hand, however, the derivatives of the logarithmic potential F log blow up at ±1 much more rapidly than F log itself, so that the classical energy estimates on F log are not enough to deduce a control on F ′ log . This problem is even more evident in the stochastic setting due to the presence in the energy estimates of the second order Itô correction, which depends of the second derivative F ′′ . Again, while this term can be handled in the case of polynomial potentials as F pol , in the case of logarithmic potential F log the situation is much more critical, due to blow-ups at ±1 pointed out above.
The second main issue is the degeneracy of the mobility. Indeed, while in the case of constant mobility m con , or more generally if m is bounded from below by a positive constant, one usually deduces estimates on ∇µ pretty directly, if m degenerates at ±1, as in the physically relevant case m pol , then there is no hope to obtain a control on the gradient of the chemical potential.
These two main problems suggest that in the case of degenerate mobility the role of the chemical potential µ must be passed by, and a different interpretation of the equation is needed. To this end, if one formally substitutes equation (1.11) into (1.10), it is possible to obtain a variational formulation on the problem only involving the variable ϕ. In particular, the nonlinear term resulting from such substitution (see Definition 2.5 below) is in the form m(ϕ)F ′′ (ϕ). Hence, supposing that the degeneracy of the mobility compensates the blow-up of F ′′ at ±1, we can obtain a coherent formulation of the problem not involving µ anymore. Such compatibility condition between m and F ′′ was employed in the mentioned work [40], and is very natural as it is satisfied by the physically relevant choices m pol and F log , as we will show in Remark 2.3 below.
The idea to overcome the presence of Itô correction terms in the energy estimates depending on F ′′ is of similar nature. If G is Lipschitz-continuous and vanishes at the extremal points ±1, its degeneracy can compensate the blow-up of F ′′ and the energy estimate can be closed. Again, such Lipschitzcontinuity assumption on G is very natural in applications, and has been widely employed in stochastic phase-field modelling. For example, in [5] the authors consider a diffuse-interface model based on the stochastic Allen-Cahn equation in the context of evolution of damage in certain continuum media. The state variable represents here the damage parameter, which may be though as the local ratio of active cohesive bonds at the microscopic level. Clearly, only positive values of the state variable are physically meaningful, and in order to ensure this the authors use a obstacle-type potential. As far as the noise is concerned, the idea to handle the singularity is of similar nature: the random forcing is supposed to be multiplicative and to "switch off" whenever the ϕ touches the potential barriers. On the same line, the superposition-type stochastic forcing that we propose in this paper has also been widely employed in applications. In [43] for instance, this has been actively explored in the context of a model for a binary mixture of incompressible fluids.
In this work we prove two main results. The first one deals with existence of martingale solutions to the problem (1.10)-(1.13) in the case of positive mobility and regular potential. Although being a preparatory work for us, this first result is also interesting on its own, as it covers the case of non constant mobility and allows up to first-order exponential growth for the potential. The proofs rely on a double approximation involving a Faedo-Galerkin discretization in space and an Yosida regularization on the nonlinearity. The second main result that we prove is the main contribution of the paper, and states existence of martingale solutions in the case of degenerate mobility and irregular potentials, possibly including F log and F ob . We employ a suitable regularisation on the potential, the mobility, and the diffusion coefficient so that we can solve the approximated problem thanks to the first result. We show then uniform estimates on the solutions by using energy and magnitude estimates based on the compatibility between m, F , and G. Finally we pass to the limit by a stochastic compactness argument.
As far as uniqueness of solutions is concerned, the problem is still open even in the deterministic setting. A uniqueness result for the system with degenerate mobility has been obtained in [49] in the framework of a nonlocal diffusion related to tumour growth dynamics: here, the authors exploit the regularizing properties of the nonlocal nature of the equation in order to show continuous dependence on the initial data. Nevertheless, for Cahn-Hilliard evolutions of local type with degenerate mobility, regularity of solutions is much more difficult to achieve, and uniqueness remains unknown. In the stochastic setting, this also prevents from proving existence of probabilistically strong solutions.
Let us now summarize the main contents of the work. In Section 2 we introduce the main setting and we state the main results. Section 3 contains the proof of existence of martingale solutions in the case of positive mobility and regular potential. Section 4 is focused on the proof of existence of martingale solutions in the setting of degenerate mobility and irregular potential.

Main results
We introduce here the notation and setting of the paper, and state the main results. The first main result focuses on existence of solutions in case of nondegenerate mobility and regular potential, while the second deals with the case of degenerate mobility and logarithmic potential.
2.1. Notation and setting. For any real Banach space E, its dual will be denoted by E * . The norm in E and the duality pairing between E * and E will be denoted by · E and ·, · E , respectively. If (A, A , ν) is a finite measure space, we use the classical notation L p (A; E) for the space of p-Bochner integrable functions, for any p ∈ [1, +∞]. We shall also use the classical symbol L 0 (A; E) for the space of A -measurable functions with values in E. If E 1 and E 2 are separable Hilbert spaces, we use the notation L 2 (E 1 , E 2 ) for the space of Hilbert-Schmidt operators from E 1 to E 2 .
Throughout the paper, (Ω, F , (F t ) t∈[0,T ] , P) is a filtered probability space satisfying the usual conditions, with T > 0 being a fixed final time, and W is a cylindrical Wiener process on a separable Hilbert space U . We fix once and for all a complete orthonormal system (u k ) k of U . For every separable Hilbert space E and ℓ ∈ [2, +∞), we set and recall that by [38,Thm. 8.20.3] we have Moreover, we will use the symbols C 0 ([0, T ]; E) and C 0 w ([0, T ]; E) for the spaces of strongly and weakly continuous functions from [0, T ] to E, respectively.
It is useful to recall here some general facts about the cylindrical Wiener process W and stochastic integration that will be used later on in the paper: we follow the approach of [63, §2.5.1-2]. Since W is cylindrical in U , we have the formal representation with (β k ) k being a family of independent real Brownian motions. Nonetheless, it is important to note that the infinite sum does not converge in general in U , hence W is not rigorously defined as a U -valued continuous process. In order to properly define W and the corresponding stochastic integral, it is useful to show that it is always possible to consider W as a Q 1 -Wiener process on a larger space U 1 , with Q 1 being of trace-class on U 1 . To this end, note that there always exists a larger separable Hilbert space U 1 and a Hilbert-Schmidt operator ι ∈ L 2 (U, U 1 ). For example, on U one can define the norm It is easy to check that · U1 is a well-defined norm on U , weaker than the usual one · U . It makes sense then to define U 1 as the abstract closure of U with respect to the norm · U1 : namely, U 1 is the abstract space of infinite linear combinations v of (u k ) k for which v U1 is finite, i.e.
With this choice, (U 1 , · U1 ) is actually a separable Hilbert space and the inclusion ι : U ֒→ U 1 is Hilbert-Schmidt: indeed, we have Consequently, by the properties of Hilbert-Schmidt operators (see again [63]), we have that the infinite sum (2.1) actually converges in U 1 : this means that we can look at W as a rigorously defined stochastic process on U 1 . Moreover, it actually holds that W is a well-defined Q 1 -Wiener process on U 1 , with Q 1 := ι • ι * being of trace class on U 1 , and such that Q In the sequel we will say that W is a cylindrical Wiener process on U if and only if it is a Q 1 -Wiener process on U 1 . Furthermore, stochastic integration with respect to the cylindrical process W is defined in terms of the usual stochastic integration with respect to the Q 1 -Wiener process. In this regard, for every where the right-hand side is the usual integral with respect to the Q 1 -Wiener process, the left-hand side is the stochastic integral with respect to the cylindrical Wiener process, and the equality is intended in the sense of indistinguishable continuous K-valued processes. The definition of stochastic integral with respect to the cylindrical Wiener process W can be shown to be independent of the specific Hilbert space U 1 and the Hilbert-Schmidt embedding ι.
be a smooth bounded domain. We shall use the notation Q := (0, T ) × O and Q t := (0, t) × O for every t ∈ (0, T ). Denoting by n the outward normal unit vector on O, we define the functional spaces endowed with their natural norms · H , · V1 , and · V2 , respectively. For |O| v, 1 for the spatial mean of v. We also define Moreover, we will use the symbol c to denote any arbitrary positive constant depending only on the data of the problem, whose value may be updated throughout the proofs. When we want to specify the dependence of c on specific quantities, we will indicate them through a subscript.

2.2.
Nondegenerate mobility and regular potential. In case of nondegenerate mobility and regular potential, we assume the following.
Let us point out that the assumption ND1 allows for the classical choice of the polynomial double-well potential F pol defined in (1.7), but also allows to consider polynomials of any orders and even first-order exponentials. Moreover, assumption ND2 allows of course for the constant mobility scenario, but also includes the case of positive nonconstant mobilities. Condition ND3 on the noise is widely employed in literature (see for example [9,43,44]), and ensures that in particular that G : H → L 2 (U, H) is Lipschitz-continuous and linearly bounded, and that the restriction G |V1 : The initial datum is assumed to be nonrandom in ND4: this is meaningful in relation to the physical interpretation of the model. A random initial datum could also be considered, but this would make the mathematical treatment too heavy in our opinion, as different estimates are based on different moments in Ω. Since this is not the main focus of the paper, we preferred to assume ϕ 0 nonrandom in order to make the treatment clearer and avoid technicalities: for an exact analysis on the moments of the initial datum we refer the reader to [77].
We precise now the definition of martingale solution in the case of nondegenerate mobility and regular potential.
The first main result of the paper deals with existence of martingale solutions in case of positive mobility and regular potential.
and the following energy inequality holds, for every t ∈ [0, T ]: If also then it holds thatφ

2.3.
Degenerate mobility and irregular potential. We deal now with a degenerate mobility m : [−1, 1] → R which vanishes at ±1 and an irregular potential F : (−1, 1) → R possibly of logarithmic or double-obstacle type. In this case we assume the following.
In particular, it is well defined the function The initial datum is nonrandom and satisfies Note that under assumption D1 the irregular component of the potential F is the convex part F 1 , which may explode at ±1. In condition D2 we assume on the other hand that the degeneracy of the mobility can only occur at ±1, and compensates the eventual blow up of F ′′ at ±1. This is expected from the point of view of application to phase-field modelling (see the Remark 2.3 below). Finally, the additional summability condition in D3 is a generalization of the classical compatibility condition between m and F ′′ to the stochastic framework. This can be interpreted as a compensation of the blow up of F ′′ and M ′′ in ±1 also by the component functions (g k ) k . Again, this condition is satisfied in several physically relevant scenarios (see Remark 2.3 below).

Remark 2.3 (Logarithmic potential).
Let us show now that the assumptions D1-D2 allow for the physically relevant case of degenerate mobility and logarithmic potential given by the natural choices m pol and F log defined in (1.8) and (1.5), respectively. Indeed, assumption D1 holds with obvious choice of F 1 and F 2 . Moreover, an elementary computation yields 1]) and also condition D2 is satisfied. With mobility m pol and potential F log , a sufficient condition for assumption D3 is that

5)
meaning essentially that the components (g k ) k are Lipschitz-continuous and vanish at the extremal points. Let us show that under (2.5) also D3 is satisfied. Indeed, for every r ∈ (−1, 1) and k ∈ N one has The computations for the terms g k M ′′ pol are entirely analogous, and D3 follows. We give now the definition of martingale solution in the case of degenerate mobility and irregular potential. The main idea is to formally substitute equation (1.11) in the variational formulation of the problem in order to remove the dependence on the variable µ. The advantage of the degeneracy of the mobility is that the resulting variational formulation makes sense thanks to assumption D2.
Remark 2.6. As we have anticipated, the variational formulation (2.6) in Definition 2.5 is formally obtained substituting the definition (1.11) of chemical potential in equation (1.10) and integrating by parts. The reason why we do so is that the chemical potential µ does not inherit enough regularity in the degenerate case. The main advantage of such substitution is that all the terms in (2.6) are still well-defined. Indeed, by D2 we have that mF ′′ ∈ C 0 ([−1, 1]), so that the third term on the left-hand side makes sense. Moreover, for every the Hölder inequality we have div(m(ϕ ′ )∇v) ∈ H, so that also the second term on the left-hand side of (2.6) makes sense.
We are now ready to state the second main result of the paper, ensuring existence of martingale solutions in the case of degenerate mobility and irregular potential. Both the cases of logarithmic and double-obstacle potential are covered.
Let us stress that the last assertion of Theorem 2.7 ensures that under (2.7) the concentrationφ is almost everywhere contained in the interior of the physically relevant domain, meaning that the contact set {|φ| = 1} has measure 0, or better said that Note that in general the degeneracy of the mobility at ±1 may prevent M to blow up at ±1, and (2.7) is not always satisfied. For example, an easy computation shows that for degenerate mobility m pol introduced in Remark 2.3 we have Hence, in such a case M pol is bounded in (−1, 1) and condition (2.7) is satisfied only if the potential F blows up at ±1. On the other hand, in case of polynomial mobility m α with α ≥ 2, condition (2.7) is always satisfied, irrespectively of the potential F . If (2.7) is not satisfied, then one can only infer that |φ| ≤ 1 almost everywhere, as it is natural to expect.

Positive mobility and regular potential
This section is devoted to the proof of Theorem 2.2. The main idea is to perform two separate approximations on the problem. The first one depends on a parameter λ > 0, and is obtained replacing the nonlinearity F with its Yosida approximation. The second one depends on the parameter n ∈ N and is a Faedo-Galerkin finite-dimensional approximation. Uniform estimates are proved first uniformly in n, when λ is fixed, and a passage to the limit as n → ∞ yields existence of approximated solutions for λ > 0 fixed. Secondly, further uniform estimates are proved uniformly in λ and a passage to the limit as λ → 0 gives existence of solutions to the original problem.
3.1. The approximation. First of all, since F ′′ ≥ −C F , the function γ : R → R defined as γ(r) := F ′ (r) + C F r, r ∈ R, is nondecreasing and continuous: hence, γ can be identified with a maximal monotone graph in R × R and satisfies γ(0) = 0. It makes sense then to introduce the Yosida approximation γ λ : R → R of γ for any λ > 0 and defineγ λ : R → [0, +∞) asγ λ (r) := r 0 γ λ (s) ds, r ∈ R. With this notation, we introduce the approximated potential as Let us recall that from the general theory on monotone analysis [2, Ch. 2] we know that the Yosida approximation γ λ is 1 λ -Lipschitz-continuous, hence also linearly bounded. Consequently, by definition ofγ λ there exists a constant c λ > 0 such thatγ(r) ≤ c λ (1 + |r| 2 ) for all r ∈ R. Also, noting that F ′ λ (r) = γ λ (r) − C F r for all r ∈ R by definition, this readily ensures that also F ′ λ is 1 λ -Lipschitzcontinuous and that, possibly renominating c λ , it holds Secondly, let (e j ) j∈N+ ⊂ V 2 and (α j ) j∈N+ be the sequences of eigenfunctions and eigenvalues of the negative Laplace operator with homogeneous Neumann conditions on O, respectively, i.e.
Then, possibly using a renormalization procedure, we can suppose that (e j ) j is a complete orthonormal system of H and an orthogonal system in V 1 . For every n ∈ N + , we define the finite dimensional space H n := span{e 1 , . . . , e n } ⊂ V 2 , endowed with the · H -norm.
We define the approximated operator G n : H n → L 2 (U, H n ) as One can check that G n is well-defined: indeed, for every v ∈ H n and every n ∈ N + , thanks to assumption A similar computations shows also that G n is Lipschitz-continuous from H n to L 2 (U, H n ).
Similarly, we define the approximated initial value Finally, for every n ∈ N + let m n := ρ n * m where (ρ n ) is a standard sequence of mollifiers. In particular, we have that We consider the approximated problem
We deduce that for every n ∈ N the approximated system (3.3)-(3.6) admits a unique solution

3.2.
Uniform estimates in n, with λ fixed. We show now that the approximated solution (ϕ λ,n , µ λ,n ) satisfy some energy estimates, independently of n, with λ > 0 being fixed.
First of all, integrating (3.3) on O and using Itô's formula yields Taking supremum in time, power ℓ/2 and expectations, thanks to the Burkholder-Davis-Gundy inequality we have Hence, by the Young inequality we infer that there exists c > 0, independent of λ and n, such that We want now to write Itô's formula for the free energy functional To this end, note that since H n ֒→ V 2 and F ′ λ is Lipschitz-continuous, Let us show now that also DE λ is Fréchet-differentiable with D 2 E λ : H n → L (H n , H * n ) given by Indeed, for every v, h, k ∈ H n we have that Now, since we have the continuous inclusion H n ֒→ L ∞ (O), by Hölder inequality we infer that where c n > 0 is the norm of the inclusion H n ֒→ L ∞ (O). Since F ′′ λ is continuous and bounded, the third factor on the right-hand side converges to 0 if k → 0 in H n by the dominated convergence theorem, This shows that DE λ is indeed Fréchet-differentiable with derivative D 2 E λ given as above. Furthermore, the derivatives DE λ and D 2 E λ are continuous and bounded on bounded subsets of H n , as it follows from the Lipschitz-continuity of F ′ λ and the continuity and boundedness of F ′′ λ . We can then apply Itô's formula to E λ (ϕ λ,n ) in the classical version of [28]. To this end, note that by (3.4) we have DE λ (ϕ λ,n ) = µ λ,n : we obtain then E λ (ϕ λ,n (t)) + Qt m n (ϕ λ,n (s, t))|∇µ λ,n (s, x)| 2 dx ds (3.9) Taking power ℓ/2 at both sides, supremum in time and then expectations yields, recalling the definition (3.8) and the assumption ND2, s 0 (µ λ,n (r), G n (ϕ λ,n (r)) dW (r)) H ℓ/2 for every t ∈ [0, T ], P-almost surely, for a certain constant c ℓ > 0 depending only on ℓ. Let us estimate the terms on the right-hand side separately. First of all, from the definition of the approximate initial value ϕ n 0 , since ϕ ∈ V 1 we have ∇ϕ n 0 H ≤ ∇ϕ 0 H . Let us focus on the stochastic integral. By the Burkholder-Davis-Gundy inequality and the estimate (3.2), we infer that for a certain constant c ℓ > 0 independent of n and λ. Thanks to assumption ND3 we have so that, consequently, we deduce that Summing and subtracting (µ λ,n ) O on the right-hand side, using the Poincaré-Wirtinger inequality and the Young inequality we deduce that is an arbitrarily large constant independent of n and λ. Let us focus now on the trace terms in Itô's formula. Since G(ϕ λ,n ) takes values in L 2 (U, V 1 ), we have for a certain constant c > 0 independent of n and λ. Taking the mean of (3.4) we get for a certain c > 0 independent of λ and n: we infer then that, possibly updating the value of c, (3.10) Let us estimate the two terms on the right-hand side. Recall that here λ > 0 is fixed. First of all, since F λ is bounded by a quadratic function by (3.1) and (ϕ n 0 ) n is bounded in H thanks to the properties of the orthogonal projection on H n , we have that for a certain c λ > 0 independent of n. Secondly, note that since |F ′′ λ | ≤ c λ for a certain c λ > 0 independent of n, by the same computations as above we have Putting this information together we deduce from (3.10) that there exists a positive constant c λ , independent of n, such that   The Gronwall lemma and the estimate (3.7) yield then, after updating the constant c λ , ϕ λ,n L ℓ (Ω;C 0 ([0,T ];V1)) ≤ c λ , (3.11) µ λ,n L ℓ/2 (Ω;L 2 (0,T ;V1)) + ∇µ λ,n L ℓ (Ω;L 2 (0,T ;H)) ≤ c λ . ≤ c λ,s .
3.3. Passage to the limit as n → ∞, with λ fixed. We perform here the passage to the limit as n → ∞, keeping λ > 0 fixed.
Let us show that the sequence of laws of (ϕ λ,n ) n is tight on C 0 ([0, T ]; H). To this end, let us recall that, sinces > 1/ℓ, by [79, Cor. 5, p. 86] we have the compact inclusion Hence, for every R > 0 the closed ball B R in L ∞ (0, T ; V 1 ) ∩ Ws ,ℓ (0, T ; V * 1 ) of radius R is compact in . Hence, we deduce that the family of laws of (ϕ λ,n , G n (ϕ λ,n ) · W, W ) n is tight on the product space for some measurable processes Note that possibly enlarging the new probability space, it is not restrictive to suppose that (Ω,F ,P) is independent of λ. Now, since F ′ λ is Lipschitz-continuous, we readily have and similarly, since G : in L p (Ω; L 2 (0, T ; L 2 (U, H))) ∀ p < ℓ .
The next step is to show that the limit processW is actually a cylindrical Wiener process in U , i.e. that is a Q 1 -Wiener process on U 1 . To this end, we introduce the filtratioñ , sinceW n is a (F λ,n,t ) t -martingale we haveẼ (W n (t) −W n (s))ψ(φ λ,n ,Ĩ λ,n ,W n ) = 0 ∀ n ∈ N .
Hence, letting n → ∞ we deduce, thanks to the strong convergences above, the continuity and boundedness of ψ, and the dominated convergence theorem, that yielding thatW is a (F λ,t ) t -martingale in U 1 . Similarly, sinceW n is a Q 1 -Wiener process on U 1 , by the dominated convergence theorem it holds that, for every h, k ∈ U 1 , from which we get that the tensor quadratic variation ofW on U 1 is Hence, by [28,Thm. 4.6] we deduce thatW is a Q 1 -Wiener process on U 1 , i.e. a cylindrical Wiener process on U , with respect to the filtration (F λ,t ) t .
Next, let us argue on the same same line forĨ λ . First of all, it is clear thatĨ λ is (F λ,t ) t -adapted andĨ λ (0) = 0, sinceĨ λ,n (0) = 0 for all n ∈ N. Secondly, for every s, t ∈ [0, T ] with s ≤ t and for every , the martingale property ofĨ λ,n and the dominated convergence theorem yield again We deduce thatĨ λ is an H-valued (F λ,t ) t -martingale. Similarly, sinceĨ λ,n = G n (φ λ,n ) ·W n , we have the tensor quadratic variation Consequently, given arbitrary h, k ∈ H, the dominated convergence theorem yields again that so that the tensor quadratic variation ofĨ λ is given by (3.14) Taking these remarks into account and settingM λ := G(φ λ ) ·W , with the information collected so far we know thatM λ andĨ λ are H-valued martingales with respect to (F λ,t ) t with tensor quadratic variations given by yielding, taking adjoints (and recalling that ι : U → U 1 is the usual inclusion), Hence, for every , for every h ∈ U 1 and k ∈ H, we have again by the dominated convergence theorem

Consequently, we have that
We are now ready to conclude: indeed, taking (3.14)-(3.16) into account, we get which yields thatĨ Eventually, testing (3.3) by arbitrary v ∈ V 1 and integrating in time yields, after letting n → ∞ and taking (3.17)

into account
Oφ for every t ∈ [0, T ],P-almost surely. Indeed, this follows directly form the convergences proved above, the fact that m n (φ λ,n ) → m(φ λ ) almost everywhere, the fact that |m n | ≤ m * , and the dominated convergence theorem. Moreover, testing (3.4) for almost every t ∈ (0, T ),P-almost surely.
Finally, if we take expectations in (3.9), noting the stochastic integral is a martingale and that P • Λ −1 n = P for every n, we obtain n (s, x))|g k (φ λ,n (s, x))| 2 dx ds for every t ∈ [0, T ]. We want to let n → ∞ using the convergences proved above. To this end, the first two terms on the left-hand side and all the terms on the right-hand side pass to the limit by weak lower semicontinuity and the dominated convergence theorem (recall that F ′′ λ is continuous and bounded). In order to pass to the limit by lower semicontinuity in the third term on the left-hand side, it is sufficient to show that (3.20) To prove this, note that since m n (φ λ,n ) → m(φ λ ) a.e. inΩ × Q, for any arbitrary fixed σ > 0, by the Severini-Egorov theorem there is a measurable set A σ ⊂Ω × Q such that |A c σ | ≤ σ and m n (φ λ,n ) → m(φ λ ) uniformly in A σ . In particular, we have that m n (φ λ, while the Hölder inequality and the boundedness of ∇μ λ,n in L 2 (Ω × Q) yields where c > 0 is independent of n and σ. Since σ and ζ are arbitrary, we infer that (3.20) holds. Hence, passing to the limit as n → ∞ yields by weak lower semicontinuity, for every t ∈ [0, T ], First of all, sinceP • Λ −1 n = P for every n ∈ N + , from (3.7) and weak lower semicontinuity it follows that for a certain c > 0 independent of λ.
Secondly, from the estimate (3.10) we infer that where again c is independent of λ. We want to let n → ∞ using weak lower semicontinuity of the norms at both sides. To this end, since ϕ n 0 → ϕ 0 in V 1 and F λ is bounded by a quadratic function by (3.1), Moreover, recalling also the strong convergences G n (φ λ,n ) → G(φ λ ) in L p (Ω; L 2 (0, T ; L 2 (U, H))) andφ λ,n →φ λ in L p (Ω; C 0 ([0, T ]; H)) for every p < ℓ, proved in the previous subsection, we have in particular that and the dominated convergence theorem yields We infer then, letting n → ∞ and using weak lower semicontinuity, that where the constant c is independent of λ. Let us bound the last two terms on the right-hand side uniformly in λ. First of all, recalling that F is a quadratic perturbation of the convex functionγ, by . Secondly, using the Hölder inequality and assumption ND3 we have Putting this information together, using the growth assumption on F ′′ in ND1, we have theñ where the constant c (possibly updated) is independent of λ. The Gronwall lemma yields then the estimates φ λ L ℓ (Ω;L ∞ (0,T ;V1)) ≤ c , (3.22) μ λ L ℓ/2 (Ω;L 2 (0,T ;V1)) + ∇μ λ L ℓ (Ω;L 2 (0,T ;H)) ≤ c .
In order to prove the energy inequality (2.3) we note that sinceP • Ξ −1 λ =P for all λ > 0, from (3.21) we infer that for every t ∈ [0, T ]. We want to let again λ → 0 and use the convergences just proved. To this end, for the second term on the left-hand side note that F λ is a quadratic perturbation of the convex function γ λ , whereγ λ (φ λ ) ≥γ(J λφλ ) and J λ := (I + λγ) −1 : R → R is the resolvent of γ. Noting that we easily infer that J λφλ →φ a.e. inΩ × Q, so that by weak lower semicontinuity and the Fatou lemmâ Moreover, for the third term on the left-hand side of (3.28), arguing exactly as in the proof of (3. 20) we have that E Qt m(φ(s, x))|∇μ(s, x)| 2 dx ds ≤ lim inf λ→0Ê Qt m(φ λ (s, x))|∇μ λ (s, x)| 2 dx ds .
Furthermore, for the second term on the right-hand side of (3.28) note that F λ (ϕ 0 ) ≤ F (ϕ 0 ). Finally, the last term on the right-hand side of (3.28) can pass to the limit by the dominated convergence theorem. Indeed, since Recalling that J λ : R → R is 1-Lipschitz continuous, it holds |J ′ λ | ≤ 1 and we deduce that where, by definition of γ we have γ ′ (J λ (φ λ )) = F ′′ (J λφλ ) + C F . From these computations and assumption ND1 it follows that The term in brackets on the right-hand side is uniformly integrable inΩ×Q because F (J λ (φ λ )) → F (φ) in L 1 (Ω×Q), hence so is the left-hand side by comparison. Hence, recalling that Letting then λ → 0 in (3.28) taking into account these remarks yields exactly, by weak lower semicontinuity, the energy inequality (2.3).

Degenerate mobility and irregular potential
This section is devoted to proving Theorem 2.7. The main idea of the proof is the following. We approximate the irregular potential F and the mobility m using a suitable regularization, depending on a parameter ε > 0, introduced in [40] in the deterministic setting. We show that the ε-approximated problem admits martingale solutions thanks to the already proved Theorem 2.7. Finally, exploiting the compatibility assumptions between F , m, and G, we prove uniform estimates on the solutions and pass to the limit by monotonicity and stochastic compactness arguments.
Let us also mention that a similar approximation in ε > 0 was used in [47], in the study of a nonlocal Cahn-Hilliard-Navier-Stokes deterministic model with degenerate mobility. Here the authors prove that the system with ε > 0 fixed admits a solution by relying on a time-discretisation argument, and then they show convergence as ε → 0. In our case, the idea is similar, but existence of solution at ε fixed is obtained using the λ-approximation of the ND case instead of the time-discretisation. This choice was meant to avoid technical time-measurability issues in the stochastic setting; nonetheless, we point out that both techniques are feasible also in the stochastic case.
We define the approximated mobility Note that by assumption D2 we have that m ε ∈ C 0 (R) with so that m ε satisfies ND2 and m ε = m on [−1 + ε, 1 − ε]. Furthermore, we define M ε as the unique function in C 2 (R) such that , r ∈ R .
In particular, note that by definition of m ε and D2 we have M ′′ ε ∈ L ∞ (R). We define the approximated operator G ε : H → L 2 (U, H) setting where, for every k ∈ N, Note that g k,ε ∈ W 1,∞ (R) for every k ∈ N and that so that G ε satisfies assumption ND3 and G ε = G on B 1−ε .

Uniform estimates in ε.
We show here uniform estimates on the approximated solutions.
First estimate. First of all, taking v = 1 in (4.1), using Itô's formula and the Burkholder-Davis-Gundy and Young inequalities as in the proof of (3.7) yields for a positive constant c independent of ε.