## Hokkaido Mathematical Journal

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

### Time regularity for aperiodic or irreducible random walks on groups

#### 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