A stochastic difference equation with stationary noise on groups

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.