Ludovic Goudenège
email: ludovic.goudenege@bretagne.ens-cachan.fr

Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection

Keywords: AMS 2000 subject classifications. 60H15, 60H07, 37L40 Cahn-Hilliard, stochastic partial differential equations, integration by parts formulae, reflection measures, invariant measures, singular nonlinearity. Résumé

On considère une équation aux dérivées partielles stochastique possédant une non-linéarité de type logarithmique (ou une puissance négative), avec une reflexion en zéro sous la contrainte de conservation de masse. L'équation, dirigée par un bruit blanc en espace et en temps, contient un double Laplacien. L'absence de principe de maximum pour le double Laplacien pose des difficultés pour l'utilisation d'une méthode classique de pénalisation, pour laquelle une importante propriété de monotonie est utilisée. Etant inspiré par les travaux de Debussche et Zambotti, on emploie une méthode basée sur les équations en dimension infinie, utilisant l'approximation par des équations regulières et la convegence des semi-groupes de transition liés aux équations régularisées. On démontre l'existence et l'unicité de solutions pour des données initiales positives, et on donne plusieurs resultats sur les mesures invariantes et les mesures de réflexion.

Introduction and main results

The Cahn-Hilliard-Cook equation is a model to describe phase separation in a binary alloy (see [START_REF] Cahn | On spinodal decomposition[END_REF], [START_REF] Cahn | Free energy of a nonuniform system. i. interfacial free energy[END_REF] and [START_REF] Cahn | Spinodal decomposition: a reprise[END_REF]) in the presence of thermal fluctuations (see [START_REF] Cook | Brownian motion in spinodal decomposition[END_REF] and [START_REF] Langer | Theory of spinodal decomposition in alloys[END_REF]). It takes the form:

     ∂ t u = - 1 2 ∆ (∆u -ψ(u)) + ξ, on Ω ⊂ R n , ∇u • ν = 0 = ∇(∆u) • ν, on ∂Ω, (0.1)
where t denotes the time variable and ∆ is the Laplace operator. Also u ∈ [-1, 1] represents the ratio between the two species and the noise term ξ accounts for the thermal fluctuations. The nonlinear term ψ has the double-logarithmic form:

ψ : u → ln 1 + u 1 -u -κu. (0.2)
The deterministic equation has been extensively studied first in the case where ψ is replaced by a polynomial function (see [START_REF] Cahn | Free energy of a nonuniform system. i. interfacial free energy[END_REF], [START_REF] Langer | Theory of spinodal decomposition in alloys[END_REF] and [START_REF] Novick-Cohen | Nonlinear aspects of the Cahn-Hilliard equation[END_REF]) and then for non smooth ψ (see [START_REF] Blowey | The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis[END_REF] and [START_REF] Debussche | On the Cahn-Hilliard equation with a logarithmic free energy[END_REF]). Furthermore, this model has been used successfully for describing phase separation phenomena, see for example the survey [START_REF] Novick-Cohen | The Cahn-Hilliard equation: mathematical and modeling perspectives[END_REF], and the references therein, or others recent results on spinodal decomposition and nucleation in [START_REF] Bates | The dynamics of nucleation for the Cahn-Hilliard equation[END_REF][START_REF] Blömker | Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation[END_REF][START_REF] Grant | Spinodal decomposition for the Cahn-Hilliard equation[END_REF][START_REF] Maier-Paape | Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. I. Probability and wavelength estimate[END_REF][START_REF] Maier-Paape | Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: nonlinear dynamics[END_REF][START_REF] Sander | Monte Carlo simulations for spinodal decomposition[END_REF][START_REF] Sander | Unexpectedly linear behavior for the Cahn-Hilliard equation[END_REF][START_REF] Wanner | Maximum norms of random sums and transient pattern formation[END_REF]. In the polynomial case, the concentration u is not constricted to remain between -1 and 1 and the logarithmic nonlinearity might seem preferable. Up to our knowledge, only the polynomial nonlinearity has been studied in the stochastic case (see [START_REF] Blömker | Spinodal decomposition for the Cahn-Hilliard-Cook equation[END_REF][START_REF] Blömker | Phase separation in stochastic Cahn-Hilliard models[END_REF][START_REF] Cardon-Weber | Cahn-Hilliard stochastic equation: existence of the solution and of its density[END_REF][START_REF] Cardon-Weber | Cahn-Hilliard stochastic equation: strict positivity of the density[END_REF][START_REF] Da Prato | Stochastic Cahn-Hilliard equation[END_REF][START_REF] Elezović | On the stochastic Cahn-Hilliard equation[END_REF]). This article is a step toward the mathematical comprehension of the full model with double-logarithmic term and noise. We consider the one dimensional case and consider a nonlinear term with only one singularity. Clearly, due to the noise, such an equation cannot have a solution, and a reflection measure should be added to the equation. Thus the right stochastic equation to study is:

     ∂ t X = - 1 2 ∆ ∆X + f (X) + η + ∂ θ Ẇ , with θ ∈ [0, 1] = Ω, ∇X • ν = 0 = ∇(∆X) • ν, on ∂Ω, (0.3)
where f is defined below, and where the measure is subject to the contact condition almost surely:

Xdη = 0. (0.4)
Stochastic partial differential equations with reflection can model the described problem or the evolution of random interfaces near a hard wall (see [START_REF] Funaki | Fluctuations for ∇φ interface model on a wall[END_REF] and [START_REF] Zambotti | Fluctuations for a ∇φ interface model with repulsion from a wall[END_REF]). For other results on fluctuations of random interfaces, see [START_REF] Giacomin | Equilibrium fluctuations for ∇φ interface model[END_REF]. For a detailled study of the contact set {(t, θ) : X(t, θ) = 0} and of the reflection measure η, see [START_REF] Dalang | Hitting properties of parabolic s.p.d.e.'s with reflection[END_REF], [START_REF] Zambotti | Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection[END_REF] and [START_REF] Zambotti | Integration by parts on δ-Bessel bridges, δ > 3 and related SPDEs[END_REF]. The equation (0.3) has been studied when no nonlinear term is taken into account in [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF]. In this paper, the authors have introduced various techniques needed to overcome the lack of comparison principle for fourth order equations. Indeed, the case of a second order equation was studied in [START_REF] Nualart | White noise driven quasilinear SPDEs with reflection[END_REF] where an extensive use of monotonicity is used, as well as in all the articles treating with the second order case. This article is in the spirit of [START_REF] Zambotti | Integration by parts on δ-Bessel bridges, δ > 3 and related SPDEs[END_REF] where a nonlinear term is taken into account for the second order equation. We study existence and uniqueness of solution for equation (0.3) with f of the form:

f (x) := f ln (x) := ln x, for all x > 0 +∞, for all x ≤ 0, (

or for α > 0:

f (x) := f α (x) := x -α , for all x > 0 +∞, for all x ≤ 0. (0.6)

Moreover we characterize the case when the measure η vanishes. Our method mixes ideas from [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF] and [START_REF] Zambotti | Integration by parts on δ-Bessel bridges, δ > 3 and related SPDEs[END_REF]. Additional difficulties are overcome, the main one being to understand how to deal with the nonlinear term. Again in [START_REF] Zambotti | Integration by parts on δ-Bessel bridges, δ > 3 and related SPDEs[END_REF], this term is not difficult to consider thanks to monotonicy arguments.

Our main results state that equations (0.3), (0.4) together with an initial condition have an unique solution (see 2.1 and 2.2). It is constructed thanks to the gradient structure of (0.3) and Strong Feller property. Furthermore, we prove that the measure η vanishes only for f described in (0.6) with α ≥ 3 (see 3.4).

1 Preliminaries

Notation

We denote by •, • the scalar product in L 2 (0, 1): for all h, k ∈ L 2 (0, 1) h, k = 1 0 h(θ)k(θ)dθ.

We denote by A the realization in L 2 (0, 1) of the Laplace operator with Neumann boundary condition, i.e.:

D(A) = Domain of A = {h ∈ W 2,2 (0, 1) : h ′ (0) = h ′ (1) = 0}
where the space W 2,2 (0, 1) is the classical Sobolev space. Below we use the notation W n,p and ||.|| W n,p to denote the Sobolev space W n,p (0, 1) and its associated norm. Remark that A is selfadjoint on L 2 (0, 1) and we have a complete orthonormal system of eigenvectors (e i ) i∈N in L 2 (0, 1) associated to the eigenvalues λ i := (-(iπ) 2 ) i∈N where we define:

e 0 (θ) = 1, e i (θ) = √ 2 cos(iπθ), for all i ∈ N * , for all θ ∈ [0, 1].

We denote by h the mean of h ∈ L 2 (0, 1):

h = 1 0 h(θ)dθ = h, e 0 .
Then we define for all c ∈ R : L 2 c = {h ∈ L 2 (0, 1) : h = c}, and L 2 = L 2 (0, 1). We remark that (-A) -1 : L 2 0 → L 2 0 is well defined. We denote by Q this operator. We can extend the definition of Q to L 2 (0, 1) (we denote this operator Q) by the formula: Qh = Q(h -h) + h, for all h ∈ L 2 (0, 1)

For γ ∈ R, we define (-A) γ by setting

(-A) γ h = +∞ i=1 (-λ i ) γ h i e i , when h = +∞ i=0 h i e i .
The domain of (-A) γ/2 is

V γ := D((-A) γ/2 ) = h = +∞ i=0 h i e i : +∞ i=1 (-λ i ) γ h 2 i < +∞ .
It is endowed with the seminorm

|h| γ = +∞ i=1 (-λ i ) γ h 2 i 1/2
, and with the norm h γ = |h| 2 γ + h2 1/2 , associated to the scalar product defined for all h, k ∈ V γ by (h, k) γ . For γ = -1, V -1 = D((-A) -1/2 ) is the completion of the space of functions h ∈ L 2 such that

Qh, h = Q(h -h) + h, h = (-A) -1 (h -h), h -h + h2 = (-A) -1/2 (h -h), (-A) -1/2 (h -h) = |h| 2 -1 + h2 < +∞.
To lighten notations, we set (•, •) := (•, •) -1 for the inner product of V -1 . The average plays an important role and we often work with functions with a fixed average c ∈ R. We define H c = {h ∈ H, h = c} for all c ∈ R. We set D(B) = W 1,2 0 (0, 1), B = ∂ ∂θ , D(B * ) = W 1,2 (0, 1) and B * = -∂ ∂θ .

We remark that BB * = -A. Finally, we denote by Π the orthogonal projector of V -1 onto H 0 .

We have:

Π : V -1 → H 0 h → h -h.
Notice that Π is also an orthogonal projector of L 2 onto L 2 0 . Moreover: -A Qh = Πh, for all h ∈ L 2 (0, 1).

( In order to solve the equation (0.3), we use a Lipschitz approximation of this equation. We denote by {f n } n∈N the sequence of Lipschitz functions which converges to the function f on (0, +∞), defined for n ∈ N by:

f n (x) := f (x + + 1/n), for all x ∈ R.
When f = f ln is the logarithmic function (0.5), we use the following positive antiderivative of

-f n = -f n ln F n (x) = F n ln (x) := (x + 1/n) ln(x + + 1/n) -x + + 1 -1/n,
for all x ∈ R, and the following positive antiderivative of -f = -f ln defined only on R + by:

F (x) = F ln (x) := x ln(x) -x + 1, for all x ∈ R + . When f = f α is the negative α-power function (0.6) with α = 1, we use the following antiderivative of -f n = -f n α F n (x) = F n α (x) := (x + + 1/n) 1-α α -1 + n α x -, for all x ∈ R,
and the following antiderivative of -f = -f α defined only on R + by:

F (x) = F α (x) := x 1-α α -1 , for all x ∈ R + .
Finally when α = 1, we use the following antiderivative of -

f n = -f n α F n (x) = F n α (x) := -ln(x + + 1/n) + nx -,
for all x ∈ R, and the following antiderivative of -f = -f α defined only on R + by:

F (x) = F α (x) := -ln x, for all x ∈ R + .
We use the notation f, f n , F, F n when the result holds both for f ln and f α . Otherwise we use

f ln , f n ln , F ln , F n ln or f α , f n α , F α , F n α .
With these notations, we rewrite (0.3) in the abstract form:

             dX = - 1 2 (A 2 X + Af (X))dt + BdW, X, η OT = 0, X(0, x) = x for x ∈ V -1 . (1.2)
Finally, in all the article, C denotes a constant which may depend on T and α and its value may change from one line to another.

The linear equation

The linear equation is given by

     dZ(t, x) = - 1 2 A 2 Z(t, x)dt + BdW, for all t ∈ [0, T ], Z(0, x) = x.
where x ∈ V -1 . We have

Z(t, x) = e -tA 2 /2 x + t 0 e -(t-s)A 2 /2 BdW s .
As easily seen this process is in C([0, +∞[; L 2 (0, 1)) (see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]). In particular, the mean of Z is constant and the law of the process Z(t, x) is the Gaussian measure:

Z(t, x) ∼ N e -tA 2 /2 x, Q t ,
where

Q t = t 0 e -sA 2 /2 BB * e -sA 2 /2 ds = (-A) -1 (I -e -tA 2 ).
If we let t → +∞, the law of Z(t, x) converges to the Gaussian measure on L 2 :

µ c := N (ce 0 , Q), where c = x.
Notice that the kernel of Q is {te 0 , t ∈ R} and µ c is concentrated on L 2 c . It is important to remark that the measure µ c is linked to the Brownian motion. Indeed, let (B θ ) θ∈[0,1] be a Brownian motion, then the law of Y c (θ) = B(θ) -B + c is µ c (see [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF]).

Lipschitz Approximation

For n ∈ N, we study for the following Lipschitz approximation of (1.2) with an initial condition

x ∈ V -1 :      dX n + 1 2 (A 2 X n + Af n (X n ))dt = BdW, X n (0, x) = x. (1.3)
We prove existence and uniqueness of solution in a suitable space for the equation (1.3). We then follow standard arguments to show existence and uniqueness of an invariant measure for the equation (1.3) with fixed n ∈ N, and the strong Feller property of the semigroup. First we have to define the definition of a weak solution to (1.3). We say X n is a mild solution of (1.3) if it is satisfied for all t ≥ 0:

X n (t, x) = Z(t, x) - t 0 Ae -(t-s)A 2 /2 f n (X n (s, x))ds. (1.4) Lemma 1.1 Fix n ∈ N, 0 < ε < 2/3 and p = 4(1 -ε).
For all x ∈ L 2 (0, 1) there exists a unique adapted process

X n ∈ C([0, T ]; V -1 ) ∩ L p ([0, T ]; L 2 (0, 1
)) solution of equation (1.4). Moreover for all t ≥ 0: X n (t, x), e 0 = x, e 0 .

(1.5)

Proof : The proof is classical and left to the reader. It is based on the following inequalities

||(-A) 1/2 e -tA 2 /2 h|| 0 ≤ C||h|| 0 t -1/4 , t > 0, h ∈ L 2 (1.6) ||Ae -tA 2 /2 h|| 0 ≤ C||h|| 0 t -1/2 , t > 0, h ∈ L 2 (1.7) ||e -tA 2 /2 h|| 0 ≤ C|h| -1 t -1/4 , t > 0, h ∈ L 2 . (1.8)
It is also standard to prove Lemma 1.2 For n ∈ N and c ∈ R, for all t > 0:

|X n (t, x) -X n (t, y)| -1 ≤ exp(-tπ 4 /2)|x -y| -1 , for all x, y ∈ L 2 c .
(1.9)

Proof : We consider for N ∈ N and x, y ∈ L 2 c the process :

S N (t, x, y) = N i=0 X n (t, x) -X n (t, y), e i e i , for all t ≥ 0. then t → S N (t, x, y) is C 1 with values in a (N + 1)-dimensional subspace of D(A) such that ΠS N = S N . Indeed ΠS N (t, x, y) = N i=1 X n (t, x) -X n (t, y), e i Πe i + X n (t, x) -X n (t, y), e 0 Πe 0 = N i=1 X n (t, x) -X n (t, y), e i e i = S N -X n (t, x) -X n (t, y), e 0 e 0 .
And by (1.5) and since x and y are in L 2 c X n (t, x) -X n (t, y), e 0 = xy, e 0 = 0.

By (1.1) we have -A Qh = Πh for all h ∈ L 2 (0, 1), then -A QS N = S N . Using the spectral behavior of A given in section 1, we have the following computation:

d dt |S N (t, x, y)| 2 -1 = d dt QS N (t, x, y), S N (t, x, y) = 2 d dt S N (t, x, y), QS N (t, x, y) = -A 2 S N (t, x, y), QS N (t, x, y) + -A(f n (X n (t, x)) -f n (X n (t, y))), QS N (t, x, y) = AS N (t, x, y), S N (t, x, y) + f n (X n (t, x)) -f n (X n (t, y)), S N (t, x, y) ≤ -π 4 |S N (t, x, y)| 2 -1 + f n (X n (t, x)) -f n (X n (t, y)), S N (t, x, y)
This differential inequality implies :

|S N (t, x, y)| 2 -1 ≤ e -tπ 4 |x -y| 2 -1 + t 0 e -(t-s)π 4 f n (X n (s, x)) -f n (X n (s, y)), S N (s, x, y) ds.
Moreover by letting N → +∞ we have

|S N (t, x, y)| 2 -1 → |X n (t, x) -X n (t, y)| 2 -1
, and since f n is monotone non-increasing we obtain

f n (X n (s, x)) -f n (X n (s, y)), S N (s, x, y) -→ N →+∞ f n (X n (s, x)) -f n (X n (s, y)), X n (s, x) -X n (s, y) ≤ 0.
Then the limit of the integral is nonpositive, and we obtain the expected inequality (1.9).

It is classical that X n ∈ C([0, T ]; V -1 ) ∩ L p ([0, T ]; L 2 (0, 1)) satisfies (1.4) if and only if it is a weak solution of (1.3) in the sense Definition 1.1 For n ∈ N, 0 < ε < 2/3 and p = 4(1ε), let x ∈ C([0, 1], R + ) with x > 0. We say that (X n (t, x)) t∈[0,T ] , defined on a stochastic basis linked to (W (t)) t∈[0,T ] , is a solution to

(1.3) on [0, T ] if : (a) almost surely X n (•, x) ∈ C([0, T ]; V -1 ) ∩ L p ([0, T ]; L 2 (0, 1)), (b)
for all h ∈ D(A 2 ) and for all 0 ≤ t ≤ T :

X n (t, x), h = x, h - t 0 X n (s, x), A 2 h ds - t 0 Ah, f n (X n (s, x)) ds - t 0
Bh, dW .

We now describe an important property of equation (1.3). It can be described as a gradient system in V -1 with a convex potential, and can be rewritten as:

     dX n - 1 2 A(-AX n + ∇U n (X n ))dt = BdW, X n (0, x) = x ∈ L 2 (0, 1), (1.10)
where ∇ denotes the gradient in the Hilbert space L 2 (0, 1), and : .11) Notice that ∇U n (x) = -f n (x) which is dissipative, then U n is a convex potential. Finally, we define the probabilty measure on L 2 c :

U n (x) := 1 0 F n (x(θ))dθ, x ∈ L 2 (0, 1). ( 1
ν n c (dx) = 1 Z n c exp(-U n (x))µ c (dx), (1.12)
where Z n c is a normalization constant. By Lemma 1.2, we easily obtain that the equation (1.3) in H c has a unique ergodic invariant measure and it is not difficult to prove that this measure is precisely ν n c . Since the potential U n is convex, we can prove that the transition semigroup is strong Feller. Let (P n,c t ) n∈N be the sequence of transition semigroup for an initial condition in

H c such that P n,c t φ(x) = E[φ(X n,c (t, x)], for all t ≥ 0, x ∈ H c , φ ∈ B b (H c ) and n ∈ N * ,
where X n,c (t, x) is the solution of the equation (1.10).

Proposition 1.1 For abitrary T > 0, there exists a constant C T > 0 such that for all φ ∈ B b (H c ), for all n ∈ N and for all t ∈ [0, T ]:

|P n,c t φ(x) -P n,c t φ(y)| ≤ √ C T √ t φ ∞ x -y -1 , for all x, y ∈ H c . (1.13)
Proof : We now consider the following process :

H 0 → H 0 x → X n,c (t, x) = X n (t, x + ce 0 ) -ce 0
which solves the following equation :

dX n,c - 1 2 A(-AX n,c + ∇U n (ce 0 + X n,c ))dt = BdW , X n,c (0, x) = x ∈ H 0 . (1.14)
This equation describes a gradient system in H 0 with non-degenerate noise and with a convex potential. We fix c > 0 and n ∈ N, and denote P n,c t the markov transition semigroup defined by :

P n,c t ψ(x) = E[ψ(X n,c (t, x))]
, for all t ≥ 0, for all x ∈ H 0 , for all ψ ∈ B b (H 0 ).

For all c ∈ R, for all x ∈ H c and ψ ∈ B b (H 0 ), if we set

φ : H c → R u → ψ(u -ce 0 ),
we have the following equality:

P n,c t ψ(x -ce 0 ) = P n,c t φ(x). (1.15)
Then for all c ∈ R, for all x ∈ H c and ψ ∈ B b (H 0 ), the following Bismut-Elworthy formula holds:

DP n,c t ψ(x -ce 0 ) • h = 1 t E ψ(X n,c (t, x -ce 0 )) t 0 (DX n,c (s, x -ce 0 ) • h), dW . (1.16)
Then by (1.14) and (1.16),

|DP n,c t φ(x) • h| 2 ≤ 1 t 2 φ 2 ∞ E t 0 DX n,c (s, x) • h 2 -1 ds . (1.17)
Let x and y be arbitrary elements in H c , then by the mean value theorem, for σ(y) ∈ [0, T ]

P n,c t φ(x) -P n,c t φ(y) = DP n,c t φ(x + σ(y)(x -y)) • (x -y). (1.18)
We use an estimate on DX n,c (s, x + σ(y)(xy)) • (xy) 2 -1 , (1.17) and (1.18), and we have the expected result for all x, y ∈ H c

|P n,c t φ(x) -P n,c t φ(y)| ≤ √ C T √ t φ ∞ x -y H .
As usual some computations below are formal and would be difficult to justify rigourously in our infinite dimensionnal setting. However the final result is easy to justify by Galerkin approximation (see [START_REF] Da Prato | Stochastic Cahn-Hilliard equation[END_REF], section 3.2).

For all n ∈ N we have a unique solution X n of (1.3). We want to know if these solutions converge to a solution of the equation (0.3). First we describe the definition of a weak solution for (0.3) : Definition 2.1 Let x ∈ C([0, 1], R + ) and x > 0. We say that (X(t, x)) t∈[0,T ] , η, W , defined on a filtered complete probability space Ω, P, F , (F t ) t∈[0,T ] , is a weak solution to (0.3) on [0, T ] for the initial condition x if:

(a) a.s.

X ∈ C(]0, T ] × [0, 1]; R + ) ∩ C([0, T ]; V -1 ) and X(0, x) = x, (b) a.s. η is a positive measure on (0, T ] × [0, 1], such that η(O δ,T ) < +∞ for all δ ∈ (0, T ],
(c) W is a cylindrical Wiener process on L 2 (0, 1),

(d) the process (X(•, x), W ) is (F t )-adapted, (e) a.s. f (X(•, x)) ∈ L 1 (O T ), (f ) for all h ∈ D(A 2
) and for all 0 < δ ≤ t ≤ T :

X(t, x), h = X(δ, x), h - t δ X(s, x), A 2 h ds - t δ Ah(θ), f (X(s, x)) ds -Ah, η O δ,t - t δ
Bh, dW , a.s., (g) a.s. the contact property holds :

supp(η) ⊂ {(t, θ) ∈ O T /X(t, x)(θ) = 0}, that is, X, η OT = 0.
Finally, a weak solution (X, η, W ) is a strong solution if the process t → X(t, x) is adapted to the filtration t → σ(W (s, .), s ∈ [0, t])

Remark 2.1 In (f ), the only term where we use the function f is well defined. Indeed, by (e) we have f (X(•, x)) ∈ L 1 (O T ) and by Sobolev embedding Ah ∈ D(A) ⊂ L ∞ (O T ). Hence the notation •, • should be interpreted as a duality between L ∞ and L 1 .

Pathwise uniqueness

We want to prove that for any pair (X i , η i , W ), i = 1, 2, of weak solutions of (0.3) defined on the same probability space with the same driving noise W and with X 1 0 = X 2 0 , we have (X 1 , η 1 ) = (X 2 , η 2 ). This pathwise uniqueness will be used in the next subsection to construct stationary strong solutions of (0.3).

Proposition 2.1 Let x ∈ C([0, 1], R + ) with x > 0. Let (X i , η i , W ), i = 1, 2 be two weak solutions of (0.3) with X 1 0 = x = X 2 0 . Then (X 1 , η 1 ) = (X 2 , η 2 ).
Proof : We use the following Lemma from [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF]. For the sake of completeness, we recall the proof.

Lemma 2.1 Let ζ be a finite signed measure on O δ,T , V ∈ C(O δ,T ) and c > 0. Suppose that: i) for all r ∈ [δ, T ], for all h ∈ C([0, 1]), such that h = 0, h, ζ Or,T = 0, ii) for all r ∈ [δ, T ], V (r, •) = c with V, ζ Or,T = 0, then ζ is the null measure. Proof : Let k ∈ C([0, 1]). Since ζ is a finite measure, by i) we obtain for all δ ≤ s ≤ t ≤ T : k, ζ Os,t = k, ζ Os,t = kζ(O s,t ), for all k ∈ C([0, 1]).
This implies ζ can be decomposed as ζ = γ ⊗ dθ, where γ is a measure on [0, T ]. By ii), we obtain:

0 = V, ζ Os,t = t s 1 0 V (s, θ)dθ dγ = cγ([s, t]).
We conclude that for all δ ≤ s ≤ t ≤ T , γ([s, t]) = 0, since c > 0. Thus ζ is the null measure.

We now prove the proposition. Let Y

(t) = X 1 (t, x) -X 2 (t, x) and ζ = η 1 -η 2 ,
Y is the solution of the following equation:

     dY = - 1 2 A 2 Y + A(f (X 1 ) -f (X 2 )) dt, Y (0) = 0.
(2.1)

We consider now the following approximation of Y :

Y N (t, .) = 1 N N n=0 n i=0 Y (t), e i e i .
Since Y is continous, then Y N converges uniformly to Y on O T . Notice that for all i ≥ 0, the process t → Y (t), e i has bounded variation, and in particular the process t → Y N (t) has bounded variation as process with values in a finite-dimensional subspace of D(A). Taking the scalar product in V -1 between Y and Y N , we obtain:

   d(Y, Y N ) = 2(Y N , dY ) = -Y N , A 2 Y + A(f (X 1 ) -f (X 2 )) + Aζ dt, (Y, Y N )(0) = 0. (2.2)
Moreover for all t ≥ 0, Y (t), e 0 = 0, so ΠY N (t) = Y N (t). So we have for all 0 < δ ≤ t ≤ T :

(Y (t), Y N (t)) = (Y (δ), Y N (δ)) + t δ AY N (s), Y (s) ds + t δ Y N (s), f (X 1 (s, x)) -f (X 2 (s, x)) ds + Y N (s), ζ O δ,t = (Y (δ), Y N (δ)) - 1 N N n=0 n i=0 (iπ) 2 t δ Y (s), e i 2 ds + Y N (s), ζ O δ,t + 1 N N n=0 n i=0 t δ Y (s, .), e i f (X 1 (s, x)) -f (X 2 (s, x)), e i ds ≤ (Y (δ), Y N (δ)) + Y N (s), ζ O δ,t + 1 N N n=0 n i=0 t δ Y (s, .), e i f (X 1 (s, x)) -f (X 2 (s, x)), e i ds. (2.3) For all s ∈ [δ, t], 1 N N n=0 n i=0 Y (s), e i f (X 1 (s, x)) -f (X 2 (s, x)), e i -Y (s), f (X 1 (s, x)) -f (X 2 (s, x)) = 1 N N n=0 n i=0 Y (s), e i e i -Y (s), f (X 1 (s, x)) -f (X 2 (s, x)) = Y N (s) -Y (s), f (X 1 (s, x)) -f (X 2 (s, x)) ≤ Y N (s) -Y (s) L ∞ ([0,1]) f (X 1 (s, x)) -f (X 2 (s, x)) L 1 ([0,1]) ,
where

• L ∞ ([0,1]) and • L 1 ([0,1]
) are the classical norm on the space [0, 1]. The latter term converges to zero since Y N (s) converges uniformly to Y (s) on [0, 1]. Taking the negative part, we have by Fatou's lemma: lim inf

N →+∞ t δ 1 N N n=0 n i=0 Y (s), e i f (X 1 (s, x)) -f (X 2 (s, x)), e i - ds ≥ t δ lim inf N →+∞ 1 N N n=0 n i=0 Y (s), e i f (X 1 (s, x)) -f (X 2 (s, x)), e i - ds = t δ Y (s), f (X 1 (s, x)) -f (X 2 (s, x)) -ds = 0,
since f is nonincreasing. Taking the limit in (2.3) as N grows to infinity, we obtain by the contact condition

Y (t) 2 -1 -Y (δ) 2 -1 ≤ Y, ζ O δ,t = -X 1 , η 2 O δ,t -X 2 , η 1 O δ,t ≤ 0.
Letting δ → 0, we have Y (t) = 0 for all t ≥ 0 and X 1 (t, x) = X 2 (t, x) for all t ≥ 0. Moreover, with the definition of a weak solution, we see that :

for all h ∈ D(A 2 ), Ah, ζ O δ,t = 0.
By density, we obtain ζ and V = X 1 = X 2 satisfy the hypothesis of Lemma 2.1, and therefore ζ is the null measure, i.e. η 1 = η 2 .

Convergence of invariants measures

Let :

K = {x ∈ L 2 (0, 1), x ≥ 0},
then we know that µ c is the law of Y c = B -B + c. We remark the following inclusion :

{B θ ∈ [-c/2, c/2], for all θ ∈ [0, 1]} ⊂ {Y c ∈ K}, therefore µ c (K) > 0 with c > 0.
Let us define U the potential associated to the function f . If f = f ln is the logarithmic function, U is defined by:

U (x) = U ln (x) :=    1 0 F ln (x(θ))dθ if x ∈ K, +∞ else.
If f = f α is the negative α-power function, U is defined by: 1-α . By Hölder inequality:

U (x) = U α (x) :=    1 0 F α (x(θ))dθ if 1 0 F α (x(θ)) dθ < +∞ and x ∈ K, +∞ else. Remark 2.2 Note that, for α < 1, F α (x(θ)) = - 1 1 -α x(θ)
1 0 |F α (x(θ))|dθ < +∞, for all x ∈ K.
We have the following result :

Proposition 2.2 For c > 0, ν n c ⇀ ν c := 1 Z c exp -U(x) ½ x∈K µ c (dx), when n → +∞,
where Z c is a normalization constant.

Proof : Let ψ ∈ C 0 b (L 2 , R). We want to prove that H ψ(x) exp(-U n (x))µ c (dx) -→ n→+∞ H ψ(x) exp(-U (x))½ x∈K µ c (dx). ( 2

.4)

Case 1 f = f ln is the logarithmic function.

We have that for a fixed x ∈ H,

exp(-U n (x)) -→ n→+∞ exp(-U (x))½ x∈K . (2.5)
Indeed, for all x / ∈ K there exists δ x > 0 small such that λ({θ ∈ [0, 1]/x(θ) ≤ -δ x }) > 0 and we have:

1 0 F n ln (x(θ))1 {x<0} dθ > 1 0 F n ln (x(θ))1 {x≤-δx} dθ > 0, for all n ≥ 1.
Then, since F n ln is nonincreasing on (-∞, 0):

0 ≤ exp(-U n ln (x)) ≤ exp - 1 0 F n ln (x(θ))1 {x≤-δx} dθ ≤ exp - 1 0 F n ln (-δ x )1 {x≤-δx} dθ ≤ exp -F n ln (-δ x )λ({x ≤ -δ x }) ≤ exp (1/n -δ x ) ln n -1 + 1/n λ({x ≤ -δ x }) .
And this latter term converges to zero as n grows to infinity. Now for x ∈ K, F n ln (x(θ)) converges to F ln (x(θ)) almost everywhere as n grows to infinity. Moreover

F n ln (x(θ)) ≤ ½ x≤1 + F 1 ln (x(θ))½ x>1
, and the right-hand side is clearly integrable. By the dominated convergence Theorem, we deduce (2.5). Since U n ln ≥ 0, (2.4) follows by dominated convergence Theorem. Case 2 f = f α is negative α-power function. For a fixed x ∈ L 2 , the potentials are increasing as n grows to infinity, we deduce:

exp(-U n α (x)) ≤ exp(-U 1 α (x)), for all n ≥ 1, for all x ∈ L 2 . (2.6)
The right-hand side is integrable on H, thus it suffices to prove that for a fixed x ∈ H,

exp(-U n α (x)) -→ n→+∞ exp(-U α (x))½ x∈K , (2.7)
where

exp(-U α (x))½ x∈K =    exp(-U α (x)) if 1 0 |F α (x(θ))| dθ < +∞ and x ∈ K, 0 else.
(2.8)

For x / ∈ K, there exists δ x > 0 small such that λ({θ ∈ [0, 1]/x(θ) ≤ -δ x }) > 0 and we have:

for all n ∈ N * , 1 0 F n α (x(θ))1 {x<0} dθ > 1 0 F n α (x(θ))1 {x≤-δx} dθ > 0. Then 0 ≤ exp -U n α (x) ≤ exp - 1 0 F n α (x(θ))1 {x≤-δx} dθ ≤ exp - 1 0 F n α (-δ x )1 {x≤-δx} dθ = exp (-F n α (-δ x ) λ ({x ≤ -δ x })) .
And this latter term converges to zero as n grows to infinity. Thus (2.7) holds.

For x ∈ K, such that

1 0 |F α (x(θ))| dθ < +∞, F n α (x(θ)) converges almost everywhere to F α (x(θ)) as n grows to infinity. Moreover F 1 α (x(θ)) ≤ F n α (x(θ)) ≤ F α (x(θ)) for all θ ∈ [0, 1],
and by the dominated convergence Theorem (2.7) holds.

If 1 0 |F α (x(θ))| dθ = +∞, necessarily α ≥ 1. For α > 1, F n α ≥ 0 and (2.7) follows from monotone convergence. If α = 1, we write 1 0 F n α (x(θ))dθ = 1 0 F n α (x(θ))½ x(θ)≤1/2 dθ + 1 0 F n α (x(θ))½ x(θ)>1/2 dθ.
The first term converges to

1 0 F α (x(θ))½ x(θ)≤1/

Existence of stationary solutions

In this section, we prove the existence of stationary solutions of equation (0.3) and that they are limits of stationary solutions of (1.3), in some suitable sense. Fix c > 0 and consider the unique (in law) stationary solution of (1.3) denote Xn c in H c . We are going to prove that the laws of Xn c weakly converge as n grows to infinity to a stationary strong solution of (0.3).

Theorem 2.1 Let c > 0 and T > 0. Almost surely Xn c converges as n grows to infinity to a process Xc in C(O T ). Moreover f ( Xc ) ∈ L 1 (O T ) almost surely, and setting

dη n = f n ( Xn c (t, θ))dtdθ -f ( Xc (t, θ))dtdθ, then ( Xn c , η n , W ) converges in law to ( Xc , η, W ) stationary strong solution of (0.3).
The proof of 2.1 requires arguments that differ significantly in the logarithmic case and in the negative α-power case. We thus have chosen to do two separated proofs. Some arguments however are similar and are not repeated.

Proof in the logarithmic case:

The proof is splitted in 4 steps. In step 1, assuming that a subsequence of Xn c converges in law. Its limit Xc is shown to satisfy f ln ( Xc ) ∈ L 1 (O T ) almost surely. Then in step 2, under the same assumption as in step 1, we prove that up to a further extraction the measures η n converges to a positive measure η and that ( Xc , η) is a weak solution in the probabilistic sense. It then remains to prove tightness of Xn c in step 3 and to use pathwise uniqueness to conclude in step 4.

Step 1.

Let us assume that (n k ) k∈N is a subsequence such that ( Xn k c ) n∈N converges in law in C(O T ) to a process Xc . By Skorohod's theorem, we can find a probability space and a sequence of processes (

V k , W k ) k∈N on that probability space such that (V k , W k ) → (V, W) in C(O T ) almost surely and (V k , W k ) has the same distribution as ( Xn k c , W) for all k ∈ N.
Notice that V ≥ 0 almost surely since for all t ≤ T the law of V (t, .) is ν c which is concentrated on K. Let now ξ k and ρ k be the following measures on O T :

dξ k := f n k ln (V k (t, θ))½ V k <1 dtdθ, and
dρ k := f n k ln (V k (t, θ))½ V k ≥1 dtdθ. Let y ∈ D(A) with ȳ = 0, taking h ∈ D(A 2
) such that y = Ah as a test function in (b) of Definition 1.1, we deduce that, for all 0 ≤ t ≤ T , y, ξ k + ρ k Ot has a limit when n → +∞. Moreover by the uniform convergence in C(O T ) of V k to V , we have

f n k ln (V k (t, θ))½ V k ≥1 -→ k→+∞ f ln (V (t, θ))½ V ≥1 , for all (t, θ) ∈ O T , (2.9)
and the convergence is uniform. We obtain for all 0 ≤ t ≤ T and for all h ∈ D(A):

h, ρ k Ot -→ k→+∞ Ot h(θ)f ln (V (s, θ))½ V ≥1 dsdθ. (2.10) Note that f ln (x)½ x≥1 is a continuous function so that f ln (V )½ V ≥1 ∈ L 1 (O T ).
Moreover, for any y ∈ D(A) with ȳ = 0, for all 0 ≤ t ≤ T , y, ξ k Ot has a limit when k → +∞.

(

Notice that almost surely:

f n k ln (V k (t, θ))½ V k <1 -→ k→+∞ +∞ if V (t, θ) ≤ 0, f ln (V (t, θ)) if V (t, θ) ∈ (0, 1].
(2.12)

Thus the limit of this term is not trivial. Let us now prove that the total mass ξ n (O T ) is bounded. We use the following Lemma whose proofs is postponed to the end of this section.

for all h ∈ D(A) such that h = 0, h, µ k OT ≤ M T (h), for all k ∈ N, (2.13
)

for all t ∈ O T , 1 0 w(t, θ)dθ = c T > 0 (2.14) and w k , µ k OT ≤ m T . (2.15)
Then there exists a constant MT such that

for all h ∈ C(O T ), h, µ k OT ≤ MT h ∞ , for all k ∈ N. (2.16)
and in particular µ k (O T ) is bounded uniformly for k ∈ N.

Let us denote by :

M T (h) = sup k∈N h, ξ k OT (2.17)
for h ∈ D(A) such that h = 0. By (2.11), we know that M T is well defined. Moreover we have

(V k ) + , ξ k OT = OT (V k (t, θ)) + f n k ln (V k (t, θ))½ V k <1 dtdθ. (2.18) Since (x) + f n k ln (x)½ x<1 is uniformly bounded in k ∈ N, there exists a positive constant m T such that (V k ) + , ξ k OT ≤ m T . (2.19)
Since V is almost surely positive, (V k ) + converges uniformly to V and V (t, .) = c T > 0 for all t ∈ [0, T ]. We use Lemma 2.2 and obtain lim sup

k→+∞ ξ k (O T ) < +∞.
Thanks to Fatou Lemma, we can write :

OT f ln (V (s, θ))½ V <1 dsdθ = OT lim inf k→+∞ f n k ln (V k (s, θ))½ V n <1 dsdθ ≤ lim inf k→+∞ OT f n k ln (V k (s, θ))½ V k <1 dsdθ (2.20) < +∞. It follows that almost surely f ln (V ) ∈ L 1 (O T ).
Step 2. We again assume that we have (n k ) k∈N a subsequence such that ( Xn k c ) k∈N converges in law to a process Xc . Again, by Skorohod's theorem, we can find a probability space and a sequence of processes (V k , W k ) k∈N such that almost surely (V k , W k ) → (V, W) in C(O T ) as k grows to infinity, and (V k , W k ) has the same distribution as ( Xn k c , W ) for all k ∈ N. By step 1, the total mass ξ k (O T ) is bounded and there exists (n km ) m∈N a sub-subsequence such that the measures

ξ km := f n km ln (V km (t, θ))½ V km <1 dtdθ converge to a measure ξ.
We denote by λ the following measure: And this is positive, thanks to (2.20). Therefore ζ is a positive measure. Taking the limit as m grows to infinity in the approximated equation, we obtain that for all h ∈ D(A 2 ) and for all 0 ≤ t ≤ T :

dλ := f ln (V (t, θ))½ V <1 dtdθ, (2.21
V (t, .), h = x, h - Ot V (s, θ)A 2 h(θ)dsdθ - Ot f ln (V (s, θ))Ah(θ)dsdθ -Ah, ζ Ot - t 0 Bh, dW .
This is the expected equation. Let us now show that the contact condition holds for (V, ζ). We prove in fact that for all β non negative:

0 ≤ V, ζ OT ≤ β. (2.22)
The key is to study the behavior of f n km ln (V km (t, θ))½ V km <1 near points (t, θ) ∈ O T such that V (t, θ) is small. Fix β > 0, there exists ε > 0 such that -T ε ln(ε) ≤ β. Let us define the following measures for all m ∈ N.

dξ m ε := f n km ln (V km (t, θ))½ V km <ε dtdθ, dτ m ε := f n km ln (V km (t, θ))½ ε≤V km <1 dtdθ, dλ ε := f ln (V (t, θ))½ V <ε dtdθ, dτ ε := f ln (V (t, θ))½ ε≤V <1 dtdθ. Clearly τ m ε converges to τ ε , it follows lim sup m→+∞ V km , ζ m OT = lim sup m→+∞ V km , ξ km ε OT -V km , λ ε OT + V km , τ m ε OT -V km , τ ε OT = lim sup m→+∞ OT V km f n km ln (V km )½ V km <ε dtdθ - OT V km f ln (V )½ V <ε dtdθ ≤ lim sup m→+∞ OT V km f n km ln (V km )½ 0≤V km <ε dtdθ + lim sup m→+∞ OT (V km ) -f ln (V )½ V <ε dtdθ
Since (V km ) -converges uniformly to zero, we deduce:

lim sup m→+∞ V km , ζ m OT ≤ T lim sup m→+∞ sup x∈[0,ε] -x ln x + 1 n km ≤ T lim sup m→+∞ -ε ln ε + 1 n km ≤ -T ε ln (ε) .
Thus the contact condition holds.

Step 3. By the convergence of the family (ν n c ) n∈N , we know that the initial distribution of Xn c converges to ν c . We now follow the same argument as in [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF], to prove for all T > 0, the laws of ( Xn c ) n∈N are tight in C(O T ). Fix n ≥ 1 and T > 0, by the Lyons-Zheng's decomposition (see theorem 5.7.1 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]), we can find M h , respectively N h , two orthogonal martingales with respect to the natural filtration of Xn c (t) 0≤t≤T

, respectively the natural filtration of Xn c (Tt)

0≤t≤T

, such that for all t ∈ [0, T ] and for all h ∈ H:

(h, Xn c (t) -Xn c (0)) = 1 2 M h t - 1 2 (N h T -N h T -t ). (2.23)
Moreover, the quadratic variations are both equal to

M h t = N h t = t Πh 2 -1 . Let u i = Q γ-1 e i for all i ≥ 1.
To simplify the notations, we denote M i and N i the martingales M ui and N ui defined in (2.23). Then we have:

E Xn c (t) -Xn c (s) 2 -γ = E +∞ i=1 ( Xn c (t) -Xn c (s), v i ) 2 -γ = E +∞ i=1 ( Xn c (t) -Xn c (0), u i ) -1 -( Xn c (s) -Xn c (0), u i ) 2 -1 = 1 4 E +∞ i=1 M i t + N i T -t -M i s -N i T -s 2 -1 = 1 4 +∞ i=1 E M i t-s + N i t-s = 1 2 +∞ i=1 |t -s| u i 2 -1 = |t -s| 2 +∞ i=1 (iπ) 2-2γ ≤ |t -s| 2 K 2 -γ ,
where K -γ is the Hilbert-Schmidt's norm of inclusion of V -1 in V -γ which is finite for γ > 3/2. So we have found a constant C > 0 such that for all t, s ∈ [0, T ]:

E Xn c (t) -Xn c (s) 2 -γ 1 2 ≤ C|t -s| 1 2 .
(2.24)

Furthermore, for 0 < δ < 1/2 and r ≥ 1, since Xn c is a stationary solution, there exists C such that for all t, s ∈ [0, T ]:

E Xn c (t) -Xn c (s) 2 W δ,r (0,1) 1 2 ≤ E Xn c (t) 2
W δ,r (0,1)

1 2 + E Xn c (s) 2 W δ,r (0,1) 1 2 ≤ 2 H y 2 W δ,r (0,1) ν n c (dy) 1 2 ≤ C H y 2 W δ,r (0,1) µ c (dy) 1 2 , (2.25)
since U n ≥ 0. And this latter term is finite.

Let κ ∈ [0, 1] and set λ = κδ -(1 -κ)γ, 1 q = κ 1 r + (1 -κ) 1 2 .
Then by interpolation for all t, s ∈ [0, T ]:

E Xn c (t) -Xn c (s) 2 W λ,q (0,1) 1 2 ≤ E Xn c (t) -Xn c (s) 2 W δ,r (0,1) κ 2 E Xn c (t) -Xn c (s) 2 -γ (1-κ) 2
.

We use (2.24), (2.25) and Sobolev embedding to conclude. Indeed, for any β ∈ (0, 1/2), we can choose δ ∈ (0, 1/2), γ > 2, r ≥ 1 and κ ∈ (0, 1) such that (λβ)q > 1. It follows that there exists C such that for all s, t ∈ [0, T ]:

E Xn c (t) -Xn c (s) 2 C β ([0,1]) 1 2 ≤ C|t -s| 1-κ 4 ,
Finally, we can conclude by the theorem 7.2 in chapter 3 of [START_REF] Ethier | Markov processes[END_REF] that the laws of ( Xn c ) n∈N are tight in C(O T ).

Step 4. We use a Lemma in [START_REF] Gyöngy | Existence of strong solutions for Itô's stochastic equations via approximations[END_REF]. This lemma allows to get the convergence of the approximated solutions in probability in any space in which these approximated solutions are tight. Lemma 2.3 Let {Z n } n≥1 be a sequence of random elements on a Polish space E endowed by its borel σ-algebra. Then {Z n } n≥1 converges in probability to an E-valued random element if and any if from every pair of subsequences {(Z n 1 k , Z n 2 k ) k≥1 , one can extract a subsequence which converges weakly to a random element supported on the diagonal {(x, y) ∈ E × E, x = y}.

For any subsequence (n k ) k∈N , we have convergence of ξ k to a finite measure ξ on O T along some sub-subsequence (k m ) m∈N . Let ξ i , i = 1..2 be two such limits. By the second step, and the uniqueness of the reflexion measure, we know ζ 1 := ξ 1λ and ζ 2 := ξ 2λ are equals. So the limit of (ξ k ) k∈N is unique, and ξ k converges to its limit ξ. Assume (n 1 k ) k∈N and (n 1 k ) k∈N are two arbitrary subsequences. In the notations of the second step and by the third step, the process Xn

1 k c , Xn 2 k c
, W is tight in a suitable space. By Skorohod's therorem, we can find a probability space and a sequence of processes (

V k 1 , V k 2 , W k ) such that (V k 1 , V k 2 , W k ) → (V 1 , V 2 , W) almost surely in C(O T ), and (V k 1 , V k 2 , W k ) as the same distribution as Xn 1 k c , Xn 2 k c
, W for all k ∈ N. In the Skorohod's space, the approximated measures respectively converge to two contact measures ζ 1 and ζ 2 . By the second step, (V 1 , ζ 1 , W) and (V 2 , ζ 2 , W) are both weak solutions of (0.3). By uniqueness, necessarily

V 1 = V 2 and ζ 1 = ζ 2 . Therefore the subsequence Xn 1 k c , η n 1 k , W , Xn 2 k c , η n 2 k , W
k∈N converges in law to a process supported on the diagonal. We use Lemma 2.3 to prove that the sequence ( Xn c , η n , W ) converges in law to ( Xc , η, W ) stationary weak solution of (0.3). Moreover by pathwise uniqueness and existence of strong solutions, we obtain that every weak solution is also a strong solution.

Proof in negative α-power case:

We again split the proof in four steps.

Step 1. Let us assume that (n k ) k∈N is a subsequence such that ( Xn k c ) n∈N converges in law in C(O T ) to a process Xc . By Skorohod's theorem, we can find a probability space and a sequence of processes (V k , W k ) k∈N on that probability space such that (V k , W k ) → (V, W) in C(O T ) almost surely and (V k , W k ) has the same distribution as ( Xn k c , W) for all k ∈ N. Notice that V ≥ 0 almost surely since for all t ≤ T the law of V (t, .) is ν c which is concentrated on K. Let now ξ k be the following measure on O T : for h ∈ D(A) such that h = 0. By (2.26), we know that M T is well defined. Therefore by Lemma 2.2 it suffices to find a function w, such that w(t, .) = c T > 0, and a sequence (w k ) k∈N such that for a positive constant m T , w k converges uniformly to w such that

dξ k := f n k α (V k (t, θ))dtdθ. Let y ∈ D(A) with ȳ = 0, taking h ∈ D(A 2
w k , ξ k OT ≤ m T . (2.28)
Denote by [START_REF] Maier-Paape | Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: nonlinear dynamics[END_REF]) holds with m T = T . As in the logarithmic case, by Fatou Lemma, it follows that almost surely f α (V ) ∈ L 1 (O T ).

w k := ((V k ) + + 1/n k ) α , it converges uniformly to w := V α . Since w k f n k α (V k (t, θ)) = 1, (2.
Step 2. We again assume that we have (n k ) k∈N a subsequence such that ( Xn k c ) k∈N converges in law to a process Xc . Again, by Skorohod's theorem, we can find a probability space and a sequence of processes (V k , W k ) k∈N such that almost surely (V k , W k ) → (V, W) in C(O T ) as k grows to infinity, and (V k , W k ) has the same distribution as ( Xn k c , W ) for all k ∈ N. By step 1, the total mass ξ k (O T ) is bounded and there exists (n km ) m∈N a sub-subsequence such that the measures

ξ km := f n km α (V km (t, θ))½ V km <1 dtdθ
converges to a measure ξ. We denote by λ the following measure:

dλ := f α (V (t, θ))dtdθ, (2.29)
and ζ m := ξ mλ. Thus ζ m converges to the measure ζ := ξλ. Thanks to Fatou Lemma, ζ is a positive measure. Taking the limit as m grows to infinity in the approximated equation, we obtain that for all h ∈ D(A 2 ) and for all 0 ≤ t ≤ T :

V (t, .), h = x, h - Ot V (s, θ)A 2 h(θ)dsdθ - Ot f α (V (s, θ))Ah(θ)dsdθ -Ah, ζ Ot - t 0 Bh, dW .
This is the expected equation. Let us now show that the contact condition holds for (V, ζ).

Case 1 : 0 ≤ α < 1.
As in the second step of the logarithmic case, fix β > 0, so there exists ε > 0 such that T ε 1-α ≤ β.

Let us define the following measures for all m ∈ N:

dξ m ε := f n km α (V km (t, θ))½ V km <ε dtdθ, dτ m ε := f n km α (V km (t, θ))½ ε≤V km dtdθ, dλ ε := f α (V (t, θ))½ V <ε dtdθ, dτ ε := f α (V (t, θ))½ ε≤V dtdθ.
Since τ m ε converges to τ ε , we have

lim sup m→+∞ V km , ζ m OT = lim sup m→+∞ V km , ξ km ε OT -V km , λ ε OT + V km , τ m ε OT -V km , τ ε OT = lim sup m→+∞ OT V km f n km α (V km )½ V km <ε dtdθ - OT V km f α (V )½ V <ε dtdθ ≤ lim sup m→+∞ OT V km f n km α (V km )½ 0≤V km <ε dtdθ + lim sup m→+∞ OT (V km ) -f α (V )½ V <ε dtdθ . It follows lim sup m→+∞ V km , ζ m OT ≤ T lim sup m→+∞ sup x∈[0,ε] x x + 1 n km α ≤ T lim sup m→+∞ ε ε + 1 n km α ≤ T ε 1-α .
Thus the contact condition holds. Case 2 : α ≥ 1. Let γ > 0, we prove that for all nonegative β, 0 ≤ V α+γ , ζ OT ≤ β and conclude that the contact condition holds by Hölder inequality. Fix β > 0, so there exists ε > 0 such that T ε γ ≤ β. Let us define the following measures for all m ∈ N:

dξ m ε := f n km α (V km (t, θ))½ V km <ε dtdθ, dτ m ε := f n km α (V km (t, θ))½ ε≤V km dtdθ, dλ ε := f α (V (t, θ))½ V <ε dtdθ, dτ ε := f α (V (t, θ))½ ε≤V dtdθ.
Since τ m ε converges to τ ε , we have lim sup

m→+∞ V km α+γ , ζ m OT = lim sup m→+∞ V km α+γ , ξ km ε OT -V km α+γ , λ ε OT + V km α+γ , τ m ε OT -V km α+γ , τ ε OT = lim sup m→+∞ OT V km α+γ f n km α (V km )½ V km <ε dtdθ - OT V km α+γ f α (V )½ V <ε dtdθ ≤ lim sup m→+∞ OT V km α+γ f n km α (V km )½ 0≤V km <ε dtdθ + lim sup m→+∞ OT V km -α+γ f α (V )½ V <ε dtdθ . It follows lim sup m→+∞ V km α+γ , ζ m OT ≤ T lim sup m→+∞ sup x∈[0,ε] x α+γ x + 1 n km α ≤ T lim sup m→+∞ ε α+γ ε + 1 n km α ≤ T ε γ .
Thus the contact condition holds.

Step 3 and step 4 are strictly identical to the logarithmic case and we do not repeat them. This ends the proof of Theorem 2.1. Now we give the proof of the Lemma 2.2.

Proof of the Lemma 2.2:

We prove this Lemma thanks to the previous Lemma 2.

1. If µ k (O T ) is bounded uniformly for k ∈ N, then the constant MT = sup k∈N µ k (O T ) (2.30) satisfies (2.16). Suppose µ k (O T ) is unbounded, then there exists k 0 ∈ N such that µ k (O T ) > 0 for all k ≥ k 0 , we denote for all k ≥ k 0 ν k := µ k /µ k (O T ).
{ν k } k≥k0 is a sequence of probability measure on O T , and we can extract a subsequence {ν km } m∈N such that there exists a probability measure ν with ν km ⇀ ν when m grows to infinity. Therefore, by the uniform convergence of w k w km , ν km OT -→ l→+∞ w, ν OT .

(2.31)

And by the uniform boundedness in (2.15), we have

w km , ν km OT ≤ m T µ km (O T ) -→ l→+∞ 0, ( 2

Convergence of the semigroup

First we state the following result which is a corollary of Theorem 2.1.

Corollary 2.1 Let c > 0.

i) There exists a continuous process (X(t, x), t ≥ 0, x ∈ K ∩ H c ) with X(0, x) = x and a set K 0 dense in K ∩ H c , such that for all x ∈ K 0 there exists a unique strong solution of equation (0.3) given by (X(t, x)) t≥0 , η x , W .

ii) The law of (X(t, x) t≥0 , η x ) is a regular conditional distribution of the law of Xc , η given

Xc (0) = x ∈ K ∩ H c .
Proof : By Theorem 2.1, we have a stationary strong solution Xc in H c , such that W and Xc (0) are independent. Conditioning Xc , η on the value of Xc (0) = x, with c = x, we obtain for ν c -almost every x a strong solution that we denote (X(t, x), η x ) for all t ≥ 0 and for all x ∈ K ∩ H c . This process is the desired process. Indeed, since the support of ν c is K ∩ H c , we have a strong solution for a dense set K 0 in K ∩ H c . Notice that all processes (X(t, x)) t≥0 with x ∈ K 0 are driven by the same noise W and are continuous with values in H. Arguing as in the proof of Lemma 2.1 we see that for all x, y ∈ K 0 , for all t ≥ 0:

X(t, x) -X(t, y) -1 ≤ x -y -1 .
Then by density, we obtain a continuous process (X(t, x)) t≥0 in H c for all x ∈ K ∩ H c .

We want to prove that for any deterministic initial condition x ∈ K ∩ H c where c > 0, there exists a strong solution of equation (0.3), necessarily unique and that the process X constructed in Corollary 2.1 is a realization of such solution. We have proved this result only for x in a dense set K 0 , but thanks to the convergence of the transition semigroup P n,c , we will be able to conclude. First we prove that the transition semigroup converges on K ∩ H c . This result is explained by the following proposition :

Proposition 2.3 Let c > 0, for all φ ∈ C b (H) and x ∈ K ∩ H c : lim n→+∞ P n,c t φ(x) = E[φ(X(t, x))] =: P c t φ(x).
(2.37)

Moreover the Markov process (X(t, x), t ≥ 0, x ∈ K ∩ H c ) is strong Feller and its transition semigroup P c is such that:

|P c t φ(x) -P c t φ(y)| ≤ φ ∞ √ t x -y H , for all x, y ∈ K ∩ H c , for all t > 0. (2.

38)

Proof : By proposition 2.1 X n is strong Feller on H c and for all φ : H c → R bounded and Borel we have :

|P n,c t φ(x) -P n,c t φ(y)| ≤ φ ∞ √ t
xy H , for all x, y ∈ K ∩ H c , for all t > 0.

(2.39)

Since (ν n c ) n≥1 is tight in H c , then there exists an increasive sequence of compact sets (J p ) p∈N in H such that: lim Note that the subsequence depends on t. Therefore, we have to prove that the limit defines a semigroup and does not depend on the chosen subsequence. By the theorem 2.1, we have for all φ, ψ ∈ C b (H) :

p→+∞ sup n≥1 ν n c (H \ J p ) = 0. (2.40) Set J := ∪ p∈N J p ∩ K. Since the support of ν c is in K ∩ H c and ν c (J) = 1, then J is dense in K ∩ H c . Fix t >
E ψ Xc (0) φ Xc (t) = lim l→+∞ E ψ Xnj l c (0) φ Xnj l c (t) = lim l→+∞ H ψ(y)E φ Xnj l c (t) Xnj l c (0) = y ν nj l c (dy) = lim l→+∞ H ψ(y)P nj l ,c t φ(y)ν nj l c (dy) = H ψ(y) Θt (y)ν c (dy).
Thus, by Corollary 2.1, we have the following equality:

E [φ (X(t, x))] = Θt (x), for ν c -almost every x. (2.44) Since E[φ(X(t, .
))] and Θt are continuous on K ∩ H c , and ν c (K ∩ H c ) = 1, the equality (2.44) is true for all x ∈ K ∩ H c . Moreover the limit does not depend on the chosen subsequence, and we obtain (2.37). Since the semigroups are equi-Lipschitz, we deduce (2.38).

Existence of solutions

We have proved that there exists a continous process X which is a strong solution of equation (0.3) for an x in a dense space. In this section, we prove existence for an initial condition in K ∩ H c with c > 0.

Theorem 2.2 Let ξ be a K-valued random value with ξ > 0 almost surely and (ξ, W ) independent, then there exists a continuous process denoted (X(t, ξ)) t≥0 and a measure η ξ such that:

(a) (X(t, ξ)) t≥0 , η ξ , W is the unique strong solution of (0.3) with X(0, ξ) = ξ almost surely.

(b) The Markov process (X(t, x), t ≥ 0, x ∈ K ∩ H c ) is continous and has P c for transition semigroup which is strong Feller on H c .

(c) For all c > 0, x ∈ K ∩ H c and 0 = t 0 < t 1 < • • • < t m , (X(t i , x), i = 1, . . . , n) is the limit in distribution of (X n (t i , x)) i=1,...,m .

(d) If ξ has distribution ν c with c > 0, then (X(t, ξ)) t≥0 is equal in distribution to Xc (t) t≥0 .

Proof : By Corollary 2.1 we have a process (X(t, x), t ≥ 0, x ∈ K ∩ H c ), such that for all x in a set K 0 dense in K ∩ H c we have a strong solution (X(t, x)) t≥0 , η x , W of (0.3) with initial condition x. By proposition 2.3, we have that the Markov process X has transition semigroup P c on H c . The strong Feller property of P c implies that for all x ∈ K ∩ H c and s > 0 the law of X(s, x) is absolutely continous with respect to the invariant measure ν c . Indeed, if ν c (Γ) = 0, then ν c (P c s (½ Γ )) = ν c (Γ) = 0. So P c s (½ Γ )(x) = 0 for ν c -almost every x and by continuity for all x ∈ K ∩ H c . Therefore almost surely X(s, x) ∈ K 0 for all s > 0 and

x ∈ K ∩ H c . Fix s > 0, denote for all θ ∈ [0, 1]: X := t → X(t + s, x), W (•, θ) := t → W (t + s, θ) -W (s, θ)),
and the measure ηx such that for all T > 0, and for all h ∈ C(O T ):

h, ηx OT := O T +s s h(t -s, θ)η x (dt, dθ) So we have a process X ∈ C([0, T ]; H)∩C(O T ) and a mesure ηx on O T which is finite on [δ, T ]×[0, 1]
for all δ ≥ 0, such that ( X(t, x)) t≥0 , ηx , W is a strong solution of (0.3) with initial condition X(s, x). By continuity X(s, x) → x in H as s → 0, so ((X(t, x)) t≥0 , η x , W ) is a strong solution of (0.3) with initial condition x in the sense of the definition 2.1. Thanks to the previous results, (b), (c) and (d) are obvious.

We have proved the existence of solution to (0.3) with a reflection measure. In [START_REF] Zambotti | Integration by parts on δ-Bessel bridges, δ > 3 and related SPDEs[END_REF], L. Zambotti uses an integration by parts formula to prove that, in some cases, the reflection measure vanishes. Moreover, L. Zambotti proves that, in some other cases, the reflection measure does not vanish. He uses the theory of the Continuous Additive Functionnals described in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]. We adapt his arguments and prove similar results for our case.

Integration by parts formula

For all φ ∈ C 1 b (H c ) we denote by ∂ h φ the directional derivative of φ along h ∈ H:

∂ h φ : x → lim t→0 1 t (φ(x + th) -φ(x)), x ∈ H.
For all φ ∈ C 1 b (H), we have:

∇φ(x), h = ∂ h φ(x).
We have the following classical result (see [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF] for details): We denote by (M, M ) two indepedant copies of the standard Brownian meander (see [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF] and [START_REF] Denisov | Random walk and the Wiener process considered from a maximum point[END_REF]), and we set for all r ∈ (0, 1):

U r (θ) :=        √ rM r -θ r , θ ∈ [0, r], √ 1 -r M θ -r 1 -r , θ ∈]r, 1]. (3.1)
The starting point is the Theorem 7.1 in [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF] where the following formula has been proved for a process Y whose the law is µ.

Theorem 3.1 For all Φ in C 1 b (H, R) and h ∈ D(A): E [∂ h Φ(Y )½ Y ∈K ] = -E Y, Ah -Y • h Φ(Y )½ Y ∈K (3.2) - 1 0 h(r) 1 2π 3 r(1 -r) E Φ(U r )e -(1/2)(Ur ) 2 dr.
We denote by p U r : R + → [0, 1] the continuous version of the density of U r . By conditioning on Y = c, we obtain:

E [∂ Πh Φ(Y c )½ Yc∈K ] = -E [ Y c , Ah Φ(Y c )½ Yc∈K ] (3.3)
- 1 0 Πh(r) p U r (c) π r(1 -r) E Φ(U r )|U r = c dr,
where Y c has been defined in the section 1. Moreover, notice that we have the following classical and easy to prove integration by parts formula for the measures (ν n c ) n∈N . For all Φ in C 1 b (H) and h ∈ D(A):

H ∂ Πh Φ dν n c = - H x, Ah Φ(x)ν n c (dx) - 1 0 Πh(r) H Φ(x)f n (x(r))ν n c (dx)dr. (3.4)
We define

γ n : x → 1 Z n c exp(-U n (x)
) for all x ∈ H, where Z n c is the constant of normalization defined in (1.12). Then γ n ∈ C 1 b (H) and for all x, h ∈ K:

∇γ n (x), h = γ n (x) ∇ log γ n (x), h = γ n (x) 1 0 h(θ)f n (x(θ))dθ. (3.5)
Let φ be in C 1 b (H). We use (3.3), with Φ = φ • γ n . So we obtain:

H ∂ Πh (φ • γ n ) dµ c = - H x, Ah φ(x)γ n (x)½ x∈K µ c (dx) (3.6) - 1 0 Πh(r) p Ur (c) π r(1 -r) E φ(U r )γ n (U r )|U r = c dr.
We compute the derivative of the product, and obtain:

H (∂ Πh φ) • γ n dµ c = - H x, Ah + ∇ log γ n (x), Πh φ(x)γ n (x)½ x∈K µ c (dx) (3.7) - 1 0 Πh(r) p Ur (c) π r(1 -r) E φ(U r )γ n (U r )|U r = c dr.
We want to let n go to infinity. We have to study the convergence of all the terms. By section 2.2, the left-hand side converges to:

H (∂ Πh φ)dν c .
Denote now by I n r the following term:

I n r := p Ur (c) π r(1 -r) E φ(U r )γ n (U r ) U r = c .
Since Z n c converges, there exists C such that for all r ∈ (0, 1):

|I n r | ≤ C p Ur (c) r(1 -r) φ ∞ J n r
where J n r is defined by:

J n r := E exp - 1 0 F n (U r (θ))dθ .
In the logarithmic case and in the negative α-power case, as in section 2.2 and by dominated convergence, we have for all r ∈ (0, 1):

lim n→+∞ J n r = E exp - 1 0 F (U r (θ))dθ . (3.8)
Therefore, in the logarithmic case and in the negative α-power case for α > 1, since |J n r | < 1, by dominated convergence, the last term in (3.7) has a limit when n grows to infinity. In the negative α-power case for α ≤ 1, since

|J n r | ≤ E exp - 1 0 F 1 α (U r (θ))dθ ,
by dominated convergence, the last term in (3.7) has a limit when n grows to infinity. Moreover, if α ≥ 3, by the law of the iterated logarithm, almost surely and for all r ∈ (0, 1):

1 0 dθ (U r (θ)) α-1 = +∞.
Thus, in this case lim

n→+∞ J n r = 0, (3.9)
and, by dominated convergence, the last term in (3.7) converges to 0. Now we use the representation described in [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF] in order to prove the convergence of the first term in the right-hand side of (3.7). Denote by S n the following

S n := - H ( x, Ah + ∇ log γ n (x), Πh ) φ(x)γ n (x)½ x∈K µ c (dx) = -E [( Y c , Ah + ∇ log γ n (Y c ), Πh ) φ(Y c )γ n (Y c )½ Yc∈K ] (3.10)
We use the following Theorem whose proof is in Appendix A in [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF]. where we set ρ :

C([0, 1]) → R, ρ(u) := 1 √ 2π exp - 1 2 u 2 .
Thanks to this Theorem, we can write:

S n = - R E ( y + B, Ah + ∇ log γ n (y + B), Πh ) ×φ(y + B)γ n (y + B)ρ(y + B)½ y+B∈K B = c -y dy
We set V r = -√ rM (1) + U r . Notice that V r is 0 at time 0, then run backwards the path of M on [0, r] and then runs the path of M on ]r, 1]. Almost surely since M > 0 on ]0, 1], then V r attains the minimum -√ rM (1) only at time r. Let (τ, M, M ) be an independent triple, such that τ has the arcsine law, then V τ has the same law as B (see [START_REF] Denisov | Random walk and the Wiener process considered from a maximum point[END_REF]). We can write:

S n = - 1 0 1 π r(1 -r) R E y + V r , Ah + f 1 n + y + V r , Πh ×φ(y + V r )γ 1 n + y + V r ρ(y + V r )½ y+Vr∈K V r = c -y dy dr = - 1 0 1 π r(1 -r) R E z - 1 n + V r , Ah + f (z + V r ) , Πh ×φ(z - 1 n + V r )γ (z + V r ) ρ(z - 1 n + V r )½ z-1 n +Vr ∈K V r = c -z + 1 n dz dr.
Now we use the proposition 3.3 which is stated in the next section 3.3. Thus, we can used Fatou Lemma to prove that for all h ∈ D(A):

1 π r(1 -r) [ f (z + V r ), Πh φ ∞ γ (z + V r ) ½ z+Vr∈K ] is integrable on Ω × R × [0, 1]
. Thus, we can used the dominated convergence Theorem to see:

lim n→+∞ S n = - 1 0 1 π r(1 -r) R E z + V r , Ah + f z + V r , Πh ×φ(z + V r )γ (z + V r ) ρ(z + V r )½ z+Vr∈K V r = c -z dz dr = - R E z + V τ , Ah + f (z + V τ ) , Πh ×φ(z + V τ )γ (z + V τ ) ρ(z + V τ )½ z+Vτ ∈K V τ = c -z dz = -E Y, Ah + f (Y ), Πh φ(Y )γ (Y ) ½ Y ∈K Y = c = -E Y c , Ah + f (Y ), Πh φ(Y c )γ (Y c ) ½ Yc∈K = - H x, Ah + f (x), Πh φ(x)ν c (dx) (3.12)
For all r ∈ (0, 1), denote Σ c r such that:

Σ c r (dω) := 1 µ c (K) p Ur (c) π r(1 -r) P U r ∈ dω|U r = c , (3.13)
thus we have the following Theorem:

Theorem 3.3 For all φ in C 1 b (H) and h ∈ D(A): H ∂ Πh φ(x)½ x∈K ν c (dx) = - H x, Ah + f (x), Πh φ(x)ν c (dx) (3.14) - 1 0
Πh(r) φγdΣ c r dr.

Moreover, for α ≥ 3, the last term vanishes.

Dirichlet forms

We now describe the Dirichlet Forms and the resolvent associated to Xn c , in order to obtain the Dirichlet Forms and the resolvent associated to Xc . The first result is the following description of the generator of Z. Let ψ h : x → exp(i(x, h) -1 ) for x ∈ H c and h ∈ D(A 2 ), then the generator of Z is such that

Lψ h (x) := d dt E[ψ h (Z(t, x))] t=0 = - 1 2 ψ h (x) i(A 2 h, x) -1 + Πh 2 -1
We define for all φ ∈ C b (H c ) the resolvent of Xn The following result is standard.

Proposition 3.1 (E n,c , Exp A (H c )) is closable in L 2 (ν n c ): we denote by (E n,c , D(E n,c )) the closure. (R n,c
λ ) λ>0 is the resolvent associated with E n,c , that is, for all λ > 0 and ψ ∈ L 2 (ν n c ), R n,c λ ψ ∈ D(E n,c ) and:

λ H R n,c λ ψφ dν n c + E n,c (R n,c λ ψ, φ) = H ψφ dν n c , for all φ ∈ D(E n,c ). ( 3

.17)

Let ψ h : x → exp(i(x, h)) for x ∈ H c and h ∈ D(A 2 ). By Itô formula

L n ψ h (x) := d dt E[ψ h ( Xn c (t, x))] t=0 = Lψ h (x) + i 2 f n (x), Πh ψ h (x). (3.18)
After an easy computation, we have (L n , Exp A (H c )) is symmetric in L 2 (ν n c ) and:

H L n φψ dν n c = - 1 2 H -A∇φ, ∇ψ dν n c , for all φ, ψ ∈ Exp A (H c ). (3.19)
Moreover we define for all φ ∈ C b (H c ) the resolvent of Xc on K ∩ H c :

R c λ φ(x) := ∞ 0 e -λt E φ Xc (t, x) dt, x ∈ K ∩ H c , λ > 0. (3.20)
We also define the symmetric bilinear form:

E c (φ, ψ) := 1 2 H -A∇φ, ∇ψ dν c , for all φ, ψ ∈ C 1 b (H). (3.21)
Proceeding as in the proposition 8.1 in [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF], we can prove that for all φ, ψ ∈ C 1 b (H c ), E n,c (φ, ψ) → E c (φ, ψ) and R n,c λ φ → R c λ φ uniformly as n grows to infinity. Let ψ ∈ C b (H c ), we can write for all h ∈ D(A 2 ):

H ψφ h dν c = lim n→+∞ H ψφ h dν n c = lim n→+∞ H R n,c λ ψ(λφ h -L n φ h )dν n c = H R c λ ψ(λφ h -Lφ h )dν c - i 2 lim n→+∞ H R n,c λ ψ(x)φ h (x) f n (x), Πh dν n c . (3.22)
Then, with the proposition 3.3 below:

H ψφ h dν c = H R c λ ψ(λφ h -Lφ h )dν c - i 2 H R c λ ψ(x)φ h (x) f (x), Πh dν c - i 2 1 0 Πh(r) H R c λ ψφ h γ dΣ c r dr = λ H R c λ ψφ h dν c - i 2 H R c λ ψφ h Ah, x dν c + 1 2 H R c λ ψφ h Πh 2 -1 dν c - i 2 H R c λ ψ(x)φ h (x) f (x), Πh dν c - i 2 1 0 Πh(r) H R c λ ψφ h γ dΣ c r dr. (3.23)
Thanks to the integration by parts formula applied to R c λ ψφ h , we have:

H ψφ h dν c = λ H R c λ ψφ h dν c + E c (R c λ ψ, φ h ). (3.24)
By linearity and by density, we obtain for all λ > 0 and ψ ∈ C b (H):

λ H R c λ ψφ dν c + E c (R c λ ψ, φ) = H ψφ dν c , for all φ ∈ D, (3.25)
where we denote D := {R c λ φ, φ ∈ C b (H c ), λ > 0}. We use classical results from [START_REF] Ma | Introduction to the theory of (nonsymmetric) Dirichlet forms[END_REF], and obtain the following proposition:

Proposition 3.2 Let c > 0. i) (E c , Exp A (H c )) is closable in L 2 (ν c ): we denote by (E c , D(E c )) the closure. ii) (E c , D(E c )) is a symmetric Dirichlet form such that Lip(H c ) ⊂ D(E c ) and E c (φ, φ) ≤ |φ| 2 Lip(Hc) . iii) (R c λ ) λ>0 is the resolvent associated with E c , that is, for all λ > 0 and ψ ∈ L 2 (ν c ), R c λ ψ ∈ D(E c ) and: λ H R c λ ψφ dν c + E c (R c λ ψ, φ) = H ψφ dν c , for all φ ∈ D(E c ). (3.26)
iv) (P c t ) t≥0 is the semigroup associated with (E c , D(E c )).

Total mass of the reflection measure

We now state and prove the proposition 3.3 used above.

Proposition 3.3 For all φ ∈ C b (H c ), for all h ∈ D(A): H f n (x), h φ(x)γ n (x)µ c (dx) (3.27)
has a limit when n grows to infinity. Moreover for all 0

< δ ≤ s ≤ t ≤ T , E [η(O s,t )] < +∞.
Proof : Denote σ n r,c the measure such that for all r ∈ [0, 1], for all c > 0: The idea is to study an integration by parts formula for the law of Y c on the path space

σ n r,c (
K := {h ∈ C([0, 1]), h(θ) ≥ 0 for all θ ∈ [0, 1/2]}.
The crucial tool is that, on this space, the processes that we consider have no more fixed mean, and we can have an integration by parts formula without the constraint of zero mean. We set for all r ∈ (0, 1/2):

T r (θ) :=          √ rM r -θ r , θ ∈ [0, r], 1 2 -r M θ -r 1 2 -r , θ ∈]r, 1/2].
Moreover we set The starting point is the Lemma B.1 in [START_REF] Debussche | Conservative stochastic Cahn-Hilliard equation with reflection[END_REF] where the following formulae have been proved.

χ : θ → ½ [0,1/2] (θ)
Lemma 3.1 For all Ψ : C([0, 1/2]) → R bounded and Borel:

E [Ψ(Y c )] = √ 32 E Ψ(b + B) exp -12(m(b + B) -c) 2 + 3 8 b 2 (3.30) = R E [Ψ(y + B)ρ(y + B)] dy, (3.31)
where we set ρ :

C([0, 1/2]) → R, ρ(u) := 12 π exp -12(m(u) -c) 2 .
Moreover, for all c > 0 and Φ ∈ C 1 b (L 2 (0, 1/2)):

E ∂ χ Φ(Y c )½ Yc∈ K = E 24 (m(Y c ) -c) Φ(Y c )½ Yc∈ K (3.32) - 1/2 0 12 π 3 r(1/2 -r) E Φ(T r )e -12(m(Tr)-c) 2 dr.
We have writen Φ(Y c ) for Φ(Y c | [0,1/2] ) with a slight abuse of notation. We set now for n ≥ 1, r ∈ (0, 1/2):

Ũ n (x) := 1/2 0 F n (x(θ))dθ, x ∈ L 2 (0, 1).
We define γn : x → exp(-Ũ n (x)) for all x ∈ H. Then γn ∈ C 1 b (L 2 (0, 1/2)) and for all x, h ∈ K:

∇γ n (x), h = γn (x) ∇ log γn (x), h = γn (x) 1/2 0 h(θ)f n (x(θ))dθ. (3.33)
Moreover we define for n ≥ 1, r ∈ (0, 1/2) and Ψ ∈ C 1 b (L 2 (0, 1/2)):

Σn,c r (Ψ) := 12

π 3 r(1/2 -r) E Ψ(T r )γ n (T r )e -12(m(Tr)-c) 2 .
Let φ be in C 1 b (L 2 (0, 1/2)). We use (3.32), with Φ = φ • γ n . So we obtain:

H ∂ χ (φ • γn )½ K dµ c = E 24 (m(Y c ) -c) φ(Y c )γ n (Y c )½ Yc∈ K - 1/2 0 Σn,c r (φ)dr. (3.34)
We compute the derivative of the product, and take φ ≡ 1, then we obtain: We also define γ : x → exp(-Ũ(x)) for all x ∈ H. Moreover we define for n ≥ 1, r ∈ (0, 1/2) and

E ∇ log γn (x), χ γn (Y c )½ Yc∈ K = E 24 (m(Y c ) -c) γn (Y c )½ Yc∈ K - 1/2 0 Σn,c r (1
Ψ ∈ C 1 b (L 2 (0, 1/2)): Σc r (Ψ) := 12
π 3 r(1/2 -r) E Ψ(T r )γ(T r )e -12(m(Tr)-c) 2 .
Finally, we denote σn r,c the measure such that for all r ∈ [0, 1], for all c > 0: σn r,c (dx) := f n (x(r)) γn (x)µ c (dx) We easily prove the following result: Lemma 3.2 For all c > 0:

lim n→+∞ 1/2 0 H dσ n r,c = E 24 (m(Y c ) -c) γ(Y c )½ Yc∈ K - 1/2 0 Σc r (1)dr.
Moreover, for α ≥ 3 the last term vanishes.

We set now for n ≥ 1:

Ũ ′ n (x) := 1 1/2 F n (x(θ))dθ = U n (x) -Ũ n (x), x ∈ L 2 (0, 1).
We also define γ,n : x → exp(-Ũ ′ n (x)) for all x ∈ H.

We notice now that we can compute explicitly the conditional distribution of

Y c given (Y c (θ), θ ∈ [0, 1/2]). Indeed, we have for all u ∈ C([0, 1/2]) and Ψ ∈ C b (L 2 (0, 1)) E [Ψ(Y c )|Y c = u on [0, 1/2]] = E[Ψ( B(c, u))],
where

B(c, u) :=      u(θ), θ ∈ [0, 1/2], u(1/2) + B θ-1/2 -12(1/2 -θ)(θ -1/2) 1/2 0 B(r)dr + m(u) -c , θ ∈]1/2, 1].
Then we have:

1/2 0 H Ψdσ n r,c dr = 1 Z n c H 1/2 0 E Ψ × f n × γ,n B(c, u) γn (u)µ c (du) dr, = 1 Z n c H 1/2 0 E Ψ × γ,n B(c, u) f n (u(r))γ n (u)µ c (du) dr, = 1 Z n c H 1/2 0 E Ψ × γ,n B(c, u) σn r,c (du).
Arguing as in the proof of section 2.2, it is easy to conclude that the limit exists, which proves (3.29) and (3.27). Recall η is the limit of dη n := f n ( Xn c (t, θ))dtdθf ( Xc (t, θ))dtdθ. We just proved that for all

δ ≤ s ≤ t ≤ T E [η(O s,t )] ≤ lim inf n→+∞ E [η n (O s,t )] ≤ lim inf n→+∞ E Os,t f n ( Xn c (u, θ))du dθ ≤ lim inf n→+∞ H Os,t f n (x(θ))du dθγ n (x)µ c (dx) = (t -s) lim inf n→+∞ H 1 0 dσ n r,c dr < +∞. (3.36)
Thus the total mass of O s,t for the reflection measure η has a finite expectation.

Reflection and Revuz measures

Theorem 3.4 For all c > 0, for all x ∈ K ∩ H c : i) For α ≥ 3, the reflection measure η x of the strong solution ((X(t, x)) t≥0 , η x , W ) vanishes. ii) For α < 3, the reflection measure η x of the strong solution ((X(t, x)) t≥0 , η x , W ) does not vanishes.

Proof : Let c > 0, x ∈ K ∩ H c , and α ≥ 3. We take the expectation of equation (0.3) for the stationnary solution. We obtain for all 0 < δ ≤ s ≤ t ≤ T , for all h ∈ D(A 2 ):

E Πk, η Os,t = -E t s Xc (u), Ak du + t s Πk(θ), f ( Xc (u)) du = (s -t) E Xc (0), Ak + Πk(θ), f ( Xc (0)) = (s -t) H x, Ak + f (x), Πk ν c (dx).
We use (3.14) with φ = 1, and prove that for all k ∈ D(A), for all 0 < δ ≤ s ≤ t ≤ T :

E Πk, η Os,t = 0. (3.38)
Now, as in Lemma 2.1, η ⊗ P can be decomposed as η ⊗ P = Γ ⊗ dθ, where Γ is a measure on [0, T ] × Ω, so we obtain that for all 0 < δ ≤ s ≤ t ≤ T , for all A ⊂ Ω:

0 = E Xc , η Os,t ½ A = Ω t s 1 0 Xc (u)(θ)dθ ½ A dΓ(u, .) = c × Γ([s, t], A).
Since c > 0, we conclude that for all 0 < δ ≤ s ≤ t ≤ T , for all A ⊂ Ω, Γ([s, t], A) = 0. Thus η ⊗ P is the null measure. Since η is a positive measure, we obtain that η is the null measure almost surely. Since the law of (X(t, x) t≥0 , η x ) is a regular conditional distribution of the law of Xc , η

given Xc (0) = x ∈ K ∩ H c , we have proved i) in the Theorem 3.4.

We consider now the logarithmic case and the negative α-power case for α < 3.

Proposition 3.4 The process {X c (t, x), t ≥ 0, x ∈ H c ∩K} is a continuous Hunt process on K with infinite life-time and strong Markov, properly associated with the Dirichlet Form E c . In particular, E c is quasi-regular.

The last assertion is a consequence of Theorem IV.5.1 in [START_REF] Ma | Introduction to the theory of (nonsymmetric) Dirichlet forms[END_REF], which describes the necessity of quasi regularity of a Dirichlet Form associated with a Markov process. We now recall the definitions of the theory of Additive Functionals of a Markov process (see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]). Consider {Y c (t, x), t ≥ 0, x ∈ H c ∩ K} a Hunt process with infinite life-time and strong Markov, properly associated with the Dirichlet Form E c . We first describe the minimum admissible filtration and the minimum completed admissible filtration. We set:

F 0 ∞ = σ{Y c (s), s ∈ [0, +∞)}, F 0 t
= σ{Y c (s), 0 ≤ s ≤ t}, for all 0 ≤ t ≤ +∞.

These filtrations are called the minimum admissible filtrations. We define E := C([0, T ], H c ), and denote by P x the law of t → Y c (t, x) on the filtered space (E, F 0 ∞ ) for all x ∈ H c ∩ K. We also define P the set of all probability measures on H c ∩ K ∩ C([0, 1]). For all m ∈ P, we define the probability measure:

P m : F 0 ∞ → R Λ →
Hc∩K∩C([0,1]) P x (Λ)m(dx).

We then denote by F m ∞ (respectively F m t ) the completion of F 0 ∞ (respectively the completion of F 0 t in F m ∞ ) with respect to P m . Finally we set (a) A t is F t -adapted for all t ≥ 0.

F ∞ = m∈P
(b) There exists Λ ∈ F ∞ with P x (Λ) = 1, for all x ∈ H c ∩ K ∩ C([0, 1]), such that θ t (Λ) ⊂ Λ for all t ≥ 0, for all ω ∈ Λ, t → A t (ω) is continuous, A 0 (ω) = 0 and for all t, s ≥ 0:

A t+s (ω) = A s (ω) + A t (θ s ω), where (θ s ) s≥0 is the time-translation semigroup on E.

Moreover, by a positive continuous additive functional (PCAF in abreviation) in the strict sense of Y c , we mean a CAF in the strict sense of Y c such that:

(c) For all ω ∈ Λ, t → A t (ω) is non-decreasing. Notice that there exists a correspondence between Revuz-measures and PCAF. We refer to Chapter 5 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] and Chapter VI in [START_REF] Ma | Introduction to the theory of (nonsymmetric) Dirichlet forms[END_REF] for all basic definitions and details. In particular the definition of a martingale additive functional (MAF in abreviation), the notion of the energy of an AF, and the quasi-sets.

X c does not satisfy suitable properties to compute Revuz-measures of PCAFs in the strict sense of X c . Thus we will use a family of process (Y δ c ) δ>0 such that:

Y δ c (t, x) = X c (t + δ, x), for all x ∈ H c ∩ K, for all t ≥ 0, for all δ > 0.

Set δ > 0. Let k ∈ D(A 2 ), set h ∈ D(A) such that Ak = h and set V : H c ∩ K → V(x) := x, k . Since the Dirichlet form (E c , D(E c )) is quasi-regular, we can apply the Fukushima decomposition (see Theorem VI.2.5 in [START_REF] Ma | Introduction to the theory of (nonsymmetric) Dirichlet forms[END_REF]). We state that there exists a MAF of finite energy M [V] and a CAF of zero energy N [V] such that for E c -quasi every x:

V(Y δ c (t, x)) -V(Y δ c (0, x)) = M [V] t + N [V]
t , t ≥ 0, P δ xa.s, (3.39)

with obvious notations for P δ x . M [V] and N [V] can be extended to CAF and MAF in the strict sense of X c , which we still denote M [V] and N [V] , such that M [V] is a P x -martingale and (3.39) holds for all x ∈ H c ∩ K. We have he following expression:

1 2 t δ Bh, dW = M [V] t (Y δ c (•, x))
, for all t ≥ 0, x ∈ K, almost surely.

Moreover N [V] is a linear combination of PCAFs in the strict sense of Y δ c such that for all t ≥ 0, x ∈ K, almost surely: And the reflection measure η x cannot be identically equal to zero.

Lemma 2 . 2

 22 Let T > 0, and {µ k } k∈N be a sequence of finite positive measures on O T . Suppose there exists {w k } k∈N a sequence of functions in C(O T ) such that w k converges uniformly to w, when k grows to infinity. Suppose also there exist a function M T : C(O T ) → R + and two nonnegative constants m T and c T such that

  ) and ζ m := ξ kmλ. Thus ζ m converges to the measure ζ := ξλ. Let u be a continuous nonnegative function on O T , we have u, ζ OT = lim m→+∞ u, ζ m OT = lim m→+∞ u, ξ km OTu, λ OT .

  ) such that y = Ah as a test function in (b) of Definition 1.1, we deduce that, for all 0 ≤ t ≤ T , y, ξ k Ot has a limit when k → +∞.(2.26)Like in the logarithmic case, we now prove that the total mass ξ n (O T ) is bounded. Let us denote by :

Theorem 3 . 2 E

 32 For all Ψ : C([0, 1]) → R bounded and Borel [Ψ(y + B)ρ(y + B)] dy,(3.11)

0 e

 0 c on H c : R n,c λ φ(x) := ∞ -λt E φ( Xn c (t, x)) dt, x ∈ H c , λ > 0. (3.15) We define Exp A (H c ) ⊂ C b (H c ) as the linear span of {cos((h, •)); sin((h, •)), h ∈ D(A 2 )}. Then we define the symmetric bilinear form: E n,c (φ, ψ) := 1 2 H -A∇φ, ∇ψ dν n c , for all φ, ψ ∈ Exp A (H). (3.16)

  dx) := f n (x(r)) γ n (x)µ c (dx)It suffices to prove that:

  θ) + u(1/2))dθ.

F

  (x(θ))dθ, x ∈ L 2 (0, 1).

  ), f ( Xc (u)) du + Ah, η Os,t = 0. (3.37) Thanks to Proposition 3.3, the expectation of each term of (3.37) is finite. So let k ∈ D(A) with k = 0, taking h ∈ D(A 2 ) such that k = Ah as a test function in(3.37), we obtain for all 0 < δ ≤ s ≤ t ≤ T , for all k ∈ D(A):

  all 0 ≤ t ≤ +∞.These filtrations are the minimum completed admissible filtrations. It is now possible to define the Additive Functionals for the Markov process Y c . To avoid useless definitions, we just recall the definition of a continuous additive functional (CAF in abreviation) in the strict sense of Y c . Definition 3.1 A family of real valued functions A := (A t ) t≥0 is called a continuous additive functionnal in the strict sense of Y c if it satisfies the following conditions:

If

  A is a linear combination of PCAFs in the strict sense of Y c , the Revuz-measure of A is a Borel signed measure m on K such that for all Φ, Ψ ∈ C b (H c ):Hc Φ(x) E +∞ 0 exp(-t)Ψ(Y c (t, x))dA t ν c (dx) = Hc E +∞ 0 exp(-t)Φ(Y c (t, x))dt Ψ(x)m(dx).

( 1 0 1 0 2 Hc 1 0 2 1 0 1 0 1 0

 11212111 X c (s, x), Ah + f (X c (s, x), Πh ) ds + 1 2 Πh(θ)η x ([δ, t], dθ) = N Ah + f (z), Πh ) ν c (dz) + 1 2Πh(r)dr γ dΣ c r .(3.40)To prove the last assertion, it suffices to remark that for all Φ,Ψ ∈ C b (H c ): t)Φ(Y δ c (t))dt x, Ah + f (x), Πh + Πh(r)dr γ dΣ c r ν c (dx).Using the same arguments, we remark that there exists a CAF in the strict sense ofY δ c whose Revuz-measure is 1 2 ( z, Ah + f (z), Πh ) ν c (dz). (3.41)Since X c is a solution of the equation (0.3) in the sense of Definition 2.1, we obtain that there exists A[V] a linear combination of PCAFs in the strict sense of X c such that:1 Πh(θ)η x ([δ, t], dθ) = A [V] t (Y δ c (•, x)), for all t ≥ 0, x ∈ K, almost surely, and its Revuz-measure is: 1 2 Πh(r)dr γ dΣ c r .(3.42)Finally, we have the following equality:Πh(θ)η x (δ + dt, dθ) ν c (dx) =

1 0 H

 10 Πh(r)γdΣ c r dr.(3.43)

  1.1) We denote by B b (H c ) the space of all Borel bounded functions and C b (H

c ) the space of continous bounded functions. We set O s,t := [s, t] × [0, 1] for s, t ∈ [0, T ] with s < t and T > 0, and O t = O 0,t for 0 ≤ t ≤ T . Given a measure ζ on O s,t and a continuous function v on O s,t , we set v, ζ Os,t := Os,t vdζ.

  Since ν is a probability measure, we deduce that (2.36) holds in fact for any h ∈ C(O T ) such that h = 0. The hypothesis of Lemma 2.1 are satisfied, and we can conclude that the measure ν is null. This is a contradiction since ν is a probability measure. Then the sequence µ k (O T ) is bounded uniformly for k ∈ N, and the constant MT in (2.30) fulfills(2.16).

	Moreover, for all h ∈ D(A) such that h = 0 h, ν km	OT -→ l→+∞	h, ν OT ,	(2.34)
	and by the uniform boundedness in (2.13), for all h ∈ D(A) such that h = 0, we have
	h, ν km	OT ≤	M T (h) µ km (O T )	-→ l→+∞	0.	(2.35)
	So that for all h ∈ D(A) such that h = 0, we have
		h, ν OT = 0.		(2.36)
							.32)
	therefore
		w, ν OT = 0.		(2.33)

  0, by (2.39), for any φ ∈ C b (H) :

		sup n∈N	( P n,c t φ ∞ + [P n,c t φ] Lip(Hc) ) < +∞.	(2.41)
	Let (n lim l→+∞	sup x∈J p	|P t nj l ,c

j ) j∈N be any sequence in N. With a diagonal procedure, by Arzelà-Ascoli Theorem, there exists (n j l ) l∈N a subsequence and a function Θ t : J → R such that:

φ(x) -Θ t (x)| = 0, for all p ∈ N. (

2

.42) By density, Θ t can be extended uniquely to a bounded Lipschitz function Θt on K ∩ H c such that Θt (x) = lim l→+∞ P nj l ,c t φ(x), for all x ∈ K ∩ H c . (2.43)