A stochastic difference equation with stationary noise on groups

Abstract

We consider the stochastic difference equation $${\eta }_k={\xi }_k\varphi ({\eta }_{k-1}),k\in \mathbb{Z}$$ on a locally compact group $G$ where ${\xi }_k$ are given $G$-valued random variables, ${\eta }_k$ are unknown $G$-valued random variables and $\varphi$ 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 $\varphi$ 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 $Kz=z\varphi (K)$ for any $z$ in the support of $\mu$. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

References in corpus (1)

• Distal actions and shifted convolution property