---
abstract: 'In this paper a stochastic equation on compact groups in discrete negative time is studied. This is closely related to Tsirelson’s stochastic differential equation, of which any solution is non-strong. How the group action reflects on the set of solutions is investigated. It is applied to generalize Yor’s result and give a necessary and sufficient condition for existence of a strong solution and for uniqueness in law.'
address:
- 'Jirô Akahori, Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga, 525-8577, Japan'
- 'Chihiro Uenishi, Cadem Corporation, Ltd., Shin-Yokohama Office, Houei-Shin-Yokohama Bldg., 2-14-9, Shin-Yokohama, Kouhoku-Ku, Yokohama, 222-0033, Japan'
- 'Kouji Yano, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan'
author:
- Jirô Akahori
- Chihiro Uenishi
- Kouji Yano
title: Stochastic equation on compact groups in discrete negative time
---

March 3, 2006

Introduction
============

In contrast with that of ordinary ones, the theory of stochastic differential equations has the distinguished notions of a [*strong solution*]{} and two uniqueness properties: [*pathwise uniqueness*]{} and [*uniqueness in law*]{}. Therefore we have the following four cases: $$\text{
\begin{tabular}{c||c|c}
                                 & unique in law   & non-unique in law \\
\hline
\hline
$ \exists $ strong solution      & {$ ({\bf C0}) 