On logarithmic Sobolev inequalities for continuous time random walks on graphs

We establish modified logarithmic Sobolev inequalities for the path distrib utions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ(d). Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and sto chastic calculus. The inequalities we prove are well adapted to describe th e tail behaviour of various functionals such as the graph distance in this setting.