Elliptic equations for measures on infinite dimensional spaces and applications

• 成果类型: Article
• 署名作者: Bogachev, VI; Röckner, M
• 署名单位: Lomonosov Moscow State University; University of Bielefeld
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/PL00008789
• 发表日期: 2001
• 页码: 445-496
• 关键词: invariant-measures; diffusion-processes; PROBABILITY-DISTRIBUTIONS; stationary solutions; gibbs measures; Finite; ergodicity; REGULARITY; martingale; uniqueness

摘要

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models.