Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's

• 成果类型: Article
• 署名作者: Goldys, B.; Maslowski, B.
• 署名单位: University of New South Wales Sydney; Czech Academy of Sciences; Institute of Mathematics of the Czech Academy of Sciences
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000800
• 发表日期: 2006
• 页码: 1451-1496
• 关键词: ornstein-uhlenbeck semigroups; sobolev inequalities; geometric ergodicity; CONVERGENCE; EQUATIONS; BEHAVIOR

摘要

A formula for the transition density of a Markov process defined by an infinite-dimensional stochastic equation is given in terms of the Ornstein-Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and V-ergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finite-dimensional stochastic equations and semilinear parabolic equations are given.