On strongly Petrovskii's parabolic SPDEs in arbitrary dimension and the stochastic Cahn-Hilliard equation

Abstract

In this paper we show that the Cahn-Hilliard stochastic SPDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. This is done proving general results on SPDEs with globally Lipschitz coefficients associated with operators on smooth domains of ${\mathbb{R}}^d$ which are parabolic in the sense of Petrovskii}, and do not necessarily define a semi-group of operators. We study the regularity of the trajectories of the solutions and the absolute continuity of the law at some given time and position.

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• Density convergence of a fully discrete finite difference method for stochastic Cahn--Hilliard equation
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• Maximum principle for quasilinear SPDE's on a bounded domain without regularity assumptions