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)


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