Weak Poincare inequalities and L-2-convergence rates of Markov semigroups

In order to describe L-2-convergence rates slower than exponential. the wea k Poincare inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare inequality can be determined by each other. Conditions for the weak Poincare inequality to h old are presented, which are easy to check and which hold in many applicati ons. The weak Poincare inequality is also studied by using isoperimetric in equalities for diffusion and jump processes. Some typical examples are give n to illustrate the general results. In particular, our results are applied to the stochastic quantization of field theory in finite Volume. Moreover, a sharp criterion of weak Poincare inequalities is presented for Poisson m easures on configuration spaces. (C) 2001 Academic Press.