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

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

2004-02-18

arxiv.org/abs/math/0402307v2

Annals of Probability 2006, Vol. 34, No. 4, 1451-1496

Mathematics

Probability

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

Scientific paper

10.1214/009117905000000800

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.

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