element.

Let $ \mu = (\mu_k , k \in -\N) $ be given and consider the set $ \sP_\mu $ of solutions of the STYE on $ G $ with the noise law $ \mu $. Since the condition $ \bP \in \sP_\mu $ is equivalent to the Markov property , the following is obvious.

\[lem: closed\] The set $ \sP_\mu $ is a closed convex subset of the linear topological space of signed measures on $ G^{-\N} $ equipped with the weak-star topology.

We denote the totality of extremal points of $ \sP_\mu $ by $ \ext $. Then Lemma \[lem: closed\] implies that $ \ext \subset \sP_\mu $ and any solution $ \bQ \in \sP_\mu $ has an integral representation $$\bQ(\cdot) = \int_{\ext} \bP(\cdot) \, \gamma(d\bP)$$ for some Borel probability measure $ \gamma $ on $ \ext $.

The following is also true.

\[lem: tail\] For $ \bP \in \sP_\mu $, the solution $ \bP $ is extremal if and only if $ \F^{\eta}_{-\infty } $ is $ \bP $-trivial.

We omit the proof, since the proof of [@Yor Theorem 1, 2)] still survives for the STYE’s.

Existence of a non-strong solution {#sec: exist}
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Since the group 