Hokkaido Mathematical Journal

Time regularity for aperiodic or irreducible random walks on groups

Nick DUNGEY

Full-text: Open access

Abstract

This paper studies time regularity for the random walk governed by a probability measure $\mu$ on a locally compact, compactly generated group $G$. If $\mu$ is eventually coset aperiodic on $G$ and satisfies certain additional conditions, we establish that the associated Markov operator $T_{\mu}$ is analytic in $L^2(G)$, that is, one has an estimate $\|(I-T_{\mu}) T_{\mu}^n \| \leq cn^{-1}$, $n\in \mathbb{N}$, in $L^2$ operator norm. Alternatively, if $\mu$ is irreducible with period $d$ and satisfies certain conditions, we show that $T_{\mu}^d$ is analytic in $L^2(G)$. To obtain these results, we develop a number of interesting algebraic and spectral properties of coset aperiodic or irreducible measures on groups.

Article information

Source
Hokkaido Math. J., Volume 37, Number 1 (2008), 19-40.

Dates
First available in Project Euclid: 21 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1253539584

Digital Object Identifier
doi:10.14492/hokmj/1253539584

Mathematical Reviews number (MathSciNet)
MR2395076

Zentralblatt MATH identifier
1143.60313

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 22D05: General properties and structure of locally compact groups

Keywords
Locally compact group probability measure convolution operator irreducible random walk

Citation

DUNGEY, Nick. Time regularity for aperiodic or irreducible random walks on groups. Hokkaido Math. J. 37 (2008), no. 1, 19--40. doi:10.14492/hokmj/1253539584. https://projecteuclid.org/euclid.hokmj/1253539584


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