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# A Stochastic Difference Equation with Stationary Noise on Groups

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http://dx.doi.org/10.4153/CJM-2011-094-6
Canad. J. Math. 64(2012), 1075-1089
Published:2011-12-23
Printed: Oct 2012
• Chandiraraj Robinson Edward Raja,
Stat-Math Unit, Indian statistical institure, 8th Mile Mysore Road, Karnataka 56059, INDIA
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## Abstract

We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), \quad k \in \mathbb Z$$ on a locally compact group $G$ where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued random variables and $\eta _k$ are unknown $G$-valued random variables. 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 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.

Keywords:  dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers

MSC Classifications:  60B15 - Probability measures on groups or semigroups, Fourier transforms, factorization 60G20 - Generalized stochastic processes

© Canadian Mathematical Society, 2013

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