Long time behavior of random walks on abelian groups
Volume 118 / 2010
Colloquium Mathematicum 118 (2010), 445-464
MSC: 60-02, 60B15, 62E10, 43A05.
DOI: 10.4064/cm118-2-6
Abstract
Let $\mathbb{G}$ be a locally compact non-compact metric group. Assuming that $\mathbb{G}$ is abelian we construct symmetric aperiodic random walks on $\mathbb{G}$ with probabilities $n \mapsto \mathbb{P} (S_{2n} \in V)$ of return to any neighborhood $V$ of the neutral element decaying at infinity almost as fast as the exponential function $n \mapsto \exp (-n)$. We also show that for some discrete groups $\mathbb{G}$, the decay of the function $n \mapsto \mathbb{P}(S_{2n} \in V)$ can be made as slow as possible by choosing appropriate aperiodic random walks $S_n$ on $\mathbb{G}$.
