Upper Bounds for Symmetric Markov Transition Functions.

Abstract

Abstract : A large number of properties which are peculiar to symmetric Markov semigroups stem from the fact that such semigroups can be analyzed simultaneously by Hilbert space techniques as well as techniques coming from maximum principle considerations. The feature of symmetric Markov semigroups in which this fact is most dramatically manifested is the central role played by the Dirichlet form. In particular, the Dirichlet form is a remarkably powerful tool with which to compare symmetric Markov semigroups. The present paper consists of a number of examples which illustrate this point. There exist tight relationships between uniform decay estimates on the semigroup and certain Sobolev like inequalities involving the Dirichlet form.