equations are defined and discussed in section \[sec: TYE\].

 

[**Acknowledgments**]{}: The authors wish to express sincere thanks to Professors Marc Yor, Freddy Delbaen, Shinzo Watanabe, and Yoichiro Takahashi for stimulating discussions and valuable comments. They also thank Gen Mano and Hidehisa Alikawa, who kindly informed them of basic theorems of the representation theory. The second author expresses his hearty thanks to Professors Hiroki Aoki and Yoshiaki Kobayashi for their kind tutorial lectures.

Definitions and preliminary facts {#sec: pre}
=================================

Let $ G $ be a compact group. We consider the following stochastic equation on $ G $ in discrete negative time: $$\eta_k = \eta_{k-1} \cdot \xi_k , 
\quad k \in -\N. 
\tag{\ref{STYE}}$$

The stochastic equation is called the [*simple Tsirelson-Yor equation*]{} on $ G $, which will be abbreviated by “STYE".

For an arbitrary sequence $ \mu = (\mu_k , k \in -\N) $ of Borel probability measures $ \mu_k $ on $ G $, we consider the equation with a general noise law $ \mu $.

Let us give the precise definition of a solution of . We denote by $ \eta = ( \eta_k , k \in -\N ) $ the coordinate mapping process: $ \eta_k(\omega)=\omega(k) $ for 