Revista Matemática Iberoamericana

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

Nick Dungey

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Abstract

The basic aim of this paper is to study asymptotic properties of the convolution powers $K^{(n)} = K*K* \cdots *K$ 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 $1 < p < \infty$, and establish Davies-Gaffney estimates in $L^2$ for the iterated operators $T^n$. 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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 23, Number 2 (2007), 587-634.

Dates
First available in Project Euclid: 26 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1190831222

Mathematical Reviews number (MathSciNet)
MR2371438

Zentralblatt MATH identifier
1130.60009

Subjects
Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G50: Sums of independent random variables; random walks 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 22D05: General properties and structure of locally compact groups 35B40: Asymptotic behavior of solutions

Keywords
locally compact group Lie group amenable group random walk probability density heat kernel gaussian estimates convolution powers

Citation

Dungey , Nick. Properties of centered random walks on locally compact groups and Lie groups. Rev. Mat. Iberoamericana 23 (2007), no. 2, 587--634. https://projecteuclid.org/euclid.rmi/1190831222


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