On Strongly Petrovskii's Parabolic SPDEs in Arbitrary Dimension and Application to the Stochastic Cahn–Hilliard Equation

In this paper we show that the Cahn–Hilliard stochastic PDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. We study the regularity of the trajectories of the solution and the absolute continuity of its law at some given time and position. This is done by showing a priori estimates which heavily depend on the specific equation, and by proving general results on stochastic and deterministic integrals involving general operators on smooth domains of ^d which are parabolic in the sense of Petrovskii, and do not necessarily define a semi-group of operators. These last estimates might be used in a more general framework.