A Stochastic Difference Equation with Stationary Noise on Groups

Chandiraraj, Robinson Edward Raja

Canadian journal of mathematics, Tome 64 (2012) no. 5, pp. 1075-1089

Résumé

We consider the stochastic difference equation ${{\eta }_{k}}\,=\,{{\xi }_{k}}\phi \left( {{\eta }_{k-1}} \right),\,\,k\,\in \,\mathbb{Z}$ 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 a 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-to-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu $ is supported on $Kz\,=\,z\phi \left( K \right)$ for any $z$ in the support of $\mu $ . We also provide a necessary and sufficient condition for the existence of solutions to the equation.

Détail

DOI : 10.4153/CJM-2011-094-6

Mots-clés : 60B15, 60G20, dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers