-\N $.

The totality of solutions of the STYE with the noise law $ \mu $ will be denoted by $ \sP_\mu $.

\[SE remark\] For a given $ \mu $, a probability measure $ \bP $ on $ G^{-\N} $ belongs to $ \sP_\mu $ if and only if the following (inhomogeneous) Markov property holds: $$\bE {\left[  \phi(\eta_k) \mid \F_{k-1}  \right]} 
= \int_{G} \phi( \eta_{k-1} \cdot g ) \mu_k(dg) , \quad k \in -\N 
\label{Markov}$$ for any bounded measurable function $ \phi $ on $ G $.

In order to give a precise meaning to the table , we need to introduce the notions of strong solution and uniqueness in law. We follow the usual terminology in the theory of stochastic differential equations.

\[Def s sol\] Let $ \mu $ be given. A solution $ \bP \in \sP_\mu $ is called [*strong*]{} if $$\F^{\eta}_k \subset \F^{\xi}_k \ \mbox{up to $ \bP $-null sets}, \quad k \in -\N .$$ If $ \bP \in \sP_\mu $ is not strong, then it is called [*non-strong*]{}[^2].

\[Def uni\] Let $ \mu $ be given. It is said that [*uniqueness in law*]{} holds if the set $ \sP_\mu $ consists of at least one 