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Spread out random walks on homogeneous spaces

Part of: Markov processes Stochastic processes Probability theory on algebraic and topological structures Noncompact transformation groups Ergodic theory

Published online by Cambridge University Press:  06 October 2020

ROLAND PROHASKA Open the ORCID record for ROLAND PROHASKA [Opens in a new window]
ROLAND PROHASKA*
Affiliation:
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092Zürich, Switzerland
*
e-mail: roland.prohaska@math.ethz.ch
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Abstract

A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

Keywords

Random walkhomogeneous spaceMarkov chainHarris recurrence

MSC classification

Primary: 37A50: Relations with probability theory and stochastic processes
Secondary: 22F30: Homogeneous spaces 60G50: Sums of independent random variables; random walks 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J05: Discrete-time Markov processes on general state spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3439 - 3473
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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