Random walks on dyadic-valued solvable matrix groups

TL;DR

This paper investigates random walks on dyadic-valued solvable matrix groups, describing their Poisson boundary and conditions for triviality, with implications for understanding their long-term behavior.

Contribution

It provides a detailed description of the Poisson boundary for these groups and identifies conditions under which the boundary is trivial, advancing the understanding of their probabilistic structure.

Findings

Poisson boundary characterized for measures with finite first moment and non-zero drifts

Boundary identified with a space of matrices with real and 2-adic entries

Conditions for boundary triviality established

Abstract

This paper is concerned with random walks on a family of dyadic-valued solvable matrix groups. A description of the Poisson boundary of these groups for probability measures of finite first moment and non-zero displacements (or drifts) is given. When non-trivial, the boundary may be identified with a space of matrices with real and 2-adic entries, depending on the values in a displacement matrix associated with the random walk. Conditions for boundary triviality are also discussed.