On logarithmic Sobolev inequalities for continuous time random walks on graphs (Q1578970)

This paper is concerned with logarithmic Sobolev inequalities and Poincaré inequalities for functionals of various continuous-time random walks on discrete sets. Finite-dimensional cylindrical functions of a continuous-time random walk on the discrete cube \(\{-1,1\}^d\) are treated by Markovian tensorization of one-dimensional inequalities. The Poincaré inequality extends to functions of paths while the logarithmic Sobolev inequalities hold only for finite-dimensional cylindrical functions. Inequalities are also obtained for functionals of continuous-time random walks on the lattice \(Z^d\), for \(d=1\) by convolution of standard Poisson measures on \(Z_+\) and \(Z_-\), and then by product tensorization for \(d\geq 2\). The treatment of Poisson random measures on \(R_+\times J\), where \(J\) is a finite set, is made possible by the Clark formula, and the (modified) logarithmic Sobolev inequalities obtained in this way apply to continuous-time random walks on graphs by considering \(J\) as a set of transitions on vertices. Deviation inequalities for functionals of the above processes are deduced from the logarithmic Sobolev inequalities.