On logarithmic Sobolev inequalities for continuous time random walks on graphs

Abstract

We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZd. Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting. © Springer-Verlag 2000.