A Stochastic Difference Equation with Stationary Noise on Groups

Abstract

AbstractWe consider the stochastic difference equation on a locally compact group G, where is an automorphism of G, ξk are given G-valued random variables and ηk are unknown G-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ξk have a common law μ and prove that if G is a distal group and is a distal automorphism of G and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space K\G for some compact subgroup K of G such that μ is supported on for any z in the support of μ. We also provide a necessary and sufficient condition for the existence of solutions to the equation.