Asymptotic properties of stochastic Cahn–Hilliard equation with singular nonlinearity and degenerate noise

Ludovic Goudenège (1) , Luigi Manca (2)

1 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We consider a stochastic partial differential equation with a logarithmic nonlinearity with singularities at 1 and −1 and a constraint of conservation of the space average. The equation, driven by a trace-class space-time noise, contains a bi-Laplacian in the drift. We obtain existence of solution for equation with polynomial approximation of the nonlinearity. Tightness of this approximated sequence of solutions is proved, leading to a limit transition semi-group. We study the asymptotic properties of this semi-group, showing the existence and uniqueness of invariant measure, asymptotic strong Feller property and topological irreducibility.

Domaines: Mathématiques [math]

Ludovic Goudenège, Luigi Manca. Asymptotic properties of stochastic Cahn–Hilliard equation with singular nonlinearity and degenerate noise. Stochastic Processes and their Applications, 2015, 125 (10), pp.3785 - 3800. ⟨10.1016/j.spa.2015.05.006⟩. ⟨hal-01092481⟩