Long time behavior of random walks on abelian groups

Alexander Bendikov¹ ; Barbara Bobikau¹

¹ Institute of Mathematics Wrocław University Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Colloquium Mathematicum, Tome 118 (2010) no. 2, pp. 445-464

Résumé

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 $\mathbb{G}$.

Détail

DOI : 10.4064/cm118-2-6

Keywords: mathbb locally compact non compact metric group assuming mathbb abelian construct symmetric aperiodic random walks mathbb probabilities mapsto mathbb return neighborhood neutral element decaying infinity almost fast exponential function mapsto exp n discrete groups mathbb decay function mapsto mathbb made slow possible choosing appropriate aperiodic random walks mathbb