Stochastic Cahn-Hilliard equation. (English) Zbl 0838.60056

Da Prato, Giuseppe; Debussche, Arnaud

Nonlinear Anal., Theory Methods Appl. 26, No. 2, 241-263 (1996).

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)

Keywords:

Cahn-Hilliard equation; Laplace operator; Wiener process; space-time white noise; existence and uniqueness of a solution; existence and uniqueness of an invariant measure

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