JournalsrmiVol. 23, No. 2pp. 587–634

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

  • Nick Dungey

    Macquarie University, Sydney, Australia
Properties of centered random walks on locally compact groups and Lie groups cover
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Abstract

The basic aim of this paper is to study asymptotic properties of the convolution powers K(n)=KKKK^{(n)} = K*K* \cdots *K of a possibly non-symmetric probability density KK on a locally compact, compactly generated group GG. If KK is centered, we show that the Markov operator TT associated with KK is analytic in Lp(G)L^p(G) for 1<p<1 < p < \infty, and establish Davies-Gaffney estimates in L2L^2 for the iterated operators TnT^n. These results enable us to obtain various Gaussian bounds on K(n)K^{(n)}. In particular, when GG 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 GG is amenable, we discover that the properties of analyticity or Davies-Gaffney estimates hold only if KK is centered.

Cite this article

Nick Dungey, Properties of centered random walks on locally compact groups and Lie groups. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 587–634

DOI 10.4171/RMI/506