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

Abstract

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.

Published at http://dx.doi.org/10.1214/009117905000000800 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Exponential Mixing for a Stochastic PDE Driven by Degenerate Noise

Cited by in corpus (8)

• Harnack Inequality and Applications for Stochastic Evolution Equations with Monotone Drifts
• Limit theorems for stationary Markov processes with L2-spectral gap
• A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs
• Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces
• Dimension-free convergence rates for gradient Langevin dynamics in RKHS
• Large deviation principle of occupation measures for Non-linear monotone SPDEs
• Quantitative spectral gaps and uniform lower bounds in the small noise limit for Markov semigroups generated by hypoelliptic stochastic differential equations
• The Ornstein Uhlenbeck Bridge and Applications to Markov Semigroups