A stochastic difference equation with stationary noise on groups

TL;DR

This paper studies a stochastic difference equation on locally compact groups with stationary noise, characterizing solutions and their structure under certain group and automorphism conditions.

Contribution

It extends previous work on the one-dimensional torus to more general groups, providing conditions for solutions and describing extremal solutions via coset spaces.

Findings

Extremal solutions correspond to points on a coset space $K\backslash G$.

Necessary and sufficient conditions for the existence of solutions are established.

Solutions are characterized when the group is pointwise distal and the automorphism is distal.

Abstract

We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), ~~~~ k \in \Z $$ on a locally compact group $G$ where $\xi _k$ are given $G$-valued random variables, $\eta _k$ are unknown $G$-valued random variables and $\phi$ is an automorphism of $G$. This equation was considered by Tsirelson and Yor on one-dimensional torus. We consider the case when $\xi _k$ have a common law $\mu$ and prove that if $G$ is a pointwise distal group and $\phi$ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu$ is supported on $Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide a necessary and sufficient condition for the existence of solutions to the equation.