Random walks on groups and discrete subordination

Abstract

Abstract Let G be a locally compact unimodular group. Let d μ = ϕ d λ be a probability measure with continuous density ϕ w.r.t. the Haar measure λ. One of the important characteristics of the random walk on G driven by μ is the probability of return at time n to a small neighborhood of the starting point, a quantity controlled by ϕ ( n ) ( e ). This paper develops the idea of discrete subordination in this context. To any Bernstein function ψ, we associate a new measure μ ψ , the ψ‐subordinate of μ, and we discuss the moment properties of μ ψ as well as the behavior of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$n\mapsto \mu _\psi ^{(n)}(e)$\end{document} .