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

Caroline Cardon-Weber (1) , Annie Millet (1, 2, 3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 SAMOS - Statistique Appliquée et MOdélisation Stochastique
3 MATISSE - UMR 8595 - Modélisation Appliquée, Trajectoires Institutionnelles et Stratégies Socio-Économiques

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.

Mots clés: Parabolic operators; Cahn-Hilliard equation; Green function; SPDEs; Malliavin calculus

Domaines: Probabilités [math.PR]

Caroline Cardon-Weber, Annie Millet. On strongly Petrovskii's parabolic SPDEs in arbitrary dimension and the stochastic Cahn-Hilliard equation. Journal of Theoretical Probability, 2004, 17, pp.1-49. ⟨10.1023/B:JOTP.0000020474.79479.fa⟩. ⟨hal-00111189⟩