$ \omega \in G^{-\N} $ and $ k \in -\N $. Set $$\xi_k = (\eta_{k-1})^{-1} \cdot \eta_k , 
\quad k \in -\N. 
\label{STYE2}$$ Let $ \F^{\eta}_k $ and $ \F^{\xi}_k $ for $ k \in -\N $ denote $$\F^{\eta}_k := \sigma (\eta_k, \eta_{k-1}, \ldots) , 
\quad k \in -\N$$ and $$\F^{\xi}_k := \sigma (\xi_k, \xi_{k-1}, \ldots) , 
\quad k \in -\N$$ respectively. It is obvious that $$\F^{\xi}_k \subset \F^{\eta}_k , 
\quad k \in -\N.$$

Since the law on $ G^{-\N} $ of the process $ \eta $ determines that of the noise $ \xi $, the following definition is reasonable.

\[Def sol\] Let $ \mu = (\mu_k , k \in -\N) $ be a sequence of Borel probability measures $ \mu_k $ on $ G $. A [*solution*]{} of the STYE on $ G $ with the noise law $ \mu $ is a probability measure $ \bP $ on $ G^{-\N} $ such that the following two statements hold:\
(i) $ \xi_k $ is independent of $ \F^{\eta}_{k-1} $ under $ \bP $, for any $ k \in -\N $.\
(ii) $ \xi_k $ is distributed as $ \mu_k $ under $ \bP $, for any $ k \in 