$}& {$ ({\bf C2}) $}\\
\hline
$ \not \exists $ strong solution & {$ ({\bf C1}) $}& {$ ({\bf C3}) $}\end{tabular}
}
\label{table}$$ The celebrated theorem of T. Yamada and S. Watanabe ([@YW]) is stated as follows: [*Pathwise uniqueness implies uniqueness in law and then any solution is strong*]{}. Except for the trivial cases where there is no solution, we may say that pathwise uniqueness implies that the case [[$ ({\bf C0}) $]{}]{} occurs. In most cases the converse is also true (See A. K. Zvonkin and N. V. Krylov [@ZK]).

B. Tsirelson ([@Tsi]) has presented a remarkable example of a stochastic differential equation stated below, which enjoys uniqueness in law property but has no strong solution: In short, the case [[$ ({\bf C1}) $]{}]{} occurs. His equation has deeply been investigated from various viewpoints by many researchers: See, for example, [@ES], [@Kall] and [@LGY].

To prove the non-existence of strong solutions, Tsirelson introduced a stochastic equation on the torus $ \T = \R / \Z $ indexed by discrete negative time: $$\eta_k = \eta_{k-1} + \xi_k , 
\quad k \in -\N, 
\label{Tsirel eq}$$ where $ \xi = ( \xi_k , k \in -\N ) $ is a Gaussian driving 