Properties of centered random walks on locally compact groups and Lie groups

2007/03/18 by Nick Dungey, Dungey, Nick
Mathematics · #22D05 #22E30 #35B40 (Secondary) #60B15 (Primary) #60G50 #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #math.GR #math.PR #msc:22D05 #msc:22E30 #msc:35B40 #msc:60B15 #msc:60G50

paper · pdf · doi:10.48550/arxiv.math/0703530

52 pages. Accepted in 2006 for publication in Revista Matematica Iberoamericana

arxiv created 2007/03/18 · arxiv updated 2009/12/01

Abstract

The basic aim of this paper is to study asymptotic properties of the convolution powers K^(n) = K * K * ... * K of a possibly non-symmetric probability density K on a locally compact, compactly generated group G. If K is centered, we show that the Markov operator T associated with K is analytic in L^p(G) for 1<p<∞, and establish Davies-Gaffney estimates in L² for the iterated operators Tⁿ. These results enable us to obtain various Gaussian bounds on K^(n). In particular, when G is a Lie group we recover and extend some estimates of Alexopoulos and of Varopoulos for convolution powers of centered densities and for the heat kernels of centered sublaplacians. Finally, in case G is amenable, we discover that the properties of analyticity or Davies-Gaffney estimates hold only if K is centered.

Related

• Local limit theorems for random walks on nilpotent Lie groups
• Locally Random Groups
• Local limit theorem for random walks on symmetric spaces
• On Random Walks in large compact Lie groups
• The central limit theorem on nilpotent Lie groups