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.
Citation
Nick DUNGEY. "Time regularity for aperiodic or irreducible random walks on groups." Hokkaido Math. J. 37 (1) 19 - 40, February 2008. https://doi.org/10.14492/hokmj/1253539584
Information