Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups

TL;DR

This paper investigates stochastic equations indexed by negative integers in compact groups, characterizing extremal solutions via infinite products of independent variables and analyzing the solution set based on the noise law.

Contribution

It introduces a novel characterization of extremal solutions for these equations using infinite products, linking solutions to the properties of the driving noise.

Findings

Extremal solutions are characterized by infinite products of independent variables.

The structure of the solution set is described in terms of the law of the noise.

Results provide insights into the properties of solutions in compact group settings.

Abstract

Stochastic equations indexed by negative integers and taking values in compact groups are studied. Extremal solutions of the equations are characterized in terms of infinite products of independent random variables. This result is applied to characterize several properties of the set of all solutions in terms of the law of the driving noise.