## Stochastic Cahn-Hilliard equation.(English)Zbl 0838.60056

This paper is concerned with the study of a stochastic Cahn-Hilliard equation of the form: $\begin{cases} dX + \bigl( \Delta^2 X - \Delta f(X) \bigr) dt = dW, \\ X(0) = x. \end{cases} \tag{1}$ Here $$\Delta$$ represents the Laplace operator on $$\mathbb{R}^n$$, $$f(u)$$ is a polynomial and $$W$$ is an infinite-dimensional Wiener process. Special attention is paid to the case when $$W$$ is a space-time white noise. Due to the strong regularizing effect of the linear part, it is possible to prove existence and uniqueness of a solution $$X(t,x)$$ to (1) when $$n = 1,2,3$$. Moreover the transition semigroup $$P_t \varphi (x) = \varphi (X(t,x))$$, $$\varphi \in C_b (H)$$, corresponding to equation (1), is studied. It is proved that it is irreducible and strong Feller. Using these properties, the existence and uniqueness of an invariant measure is proved.
Reviewer: G.Da Prato (Pisa)

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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### References:

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