  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.

Probability;Dynamical Systems