- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:44
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
23
2007
2
Properties of centered random walks on locally compact groups and Lie groups
Nick
Dungey
Macquarie University, SYDNEY, NSW, AUSTRALIA
Locally compact group, Lie group, amenable group, random walk, probability density, heat kernel, Gaussian estimates, convolution powers
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.
Probability theory and stochastic processes
Topological groups, Lie groups
Partial differential equations
General
587
634
10.4171/RMI/506
http://www.ems-ph.org/doi/10.4171/RMI/506