noise and $ \eta = ( \eta_k , k \in -\N ) $ is an unknown process. We can find its simpler proof, mainly due to N. V. Krylov, in the literature: For example, see [@LS pp.150–151], [@SY pp.149–151] and [@IW pp.195–197]. In these contexts, the authors discussed, instead of , a modified equation on the real line $ \R $ as follows: $$\begin{aligned}
\eta_k = \alpha (\eta_{k-1})  + \xi_k , 
\quad k \in -\N. 
\label{Yor eq}\end{aligned}$$ Here $ \alpha (x) $ denotes the fractional part of $ x \in \R $.

M. Yor ([@Yor]) has studied the stochastic equation for an [*arbitrarily*]{} given noise law, which is not necessarily Gaussian. In this case we always have a non-strong solution, so that the case [[$ ({\bf C0}) $]{}]{} never occurs. He successfully obtained striking results to give a necessary and sufficient condition for the trichotomy [[$ ({\bf C1}) $]{}]{}-[[$ ({\bf C3}) $]{}]{} in terms of the noise law. One of his results may roughly be stated as follows. To any given noise law $ \mu $ there corresponds a subgroup $ \Z_\mu $ of $ \Z $ such that the following holds.

\[Intro YorTh\] [(i)]{} The case [[$ ({\bf C1}) 