On a Class of Stochastic Semilinear PDEs

• Published in: Stochastic analysis and applications Vol. 24; no. 2; pp. 399 - 426
• Main Author: Manca, Luigi
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 01-05-2006; Taylor & Francis
• Subjects: Differential stochastic equation; Exact sciences and technology; Invariant measure; Kolmogorov equation; Log-Sobolev inequality; Mathematical analysis; Mathematics; Partial differential equations; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Spectral gap; Stochastic analysis; Semi linear equation; Existence of solution; Invariant measure; Transition semigroup; Lipschitz function; Kolmogorov equation; Partial differential equation; Stochastic equation
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1080/07362990500522452

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L
2
(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.