the underlying probability space with respect to $\widetilde{\Omega} = \{\tau_x < +\infty\}$. The process $X_{\tau_x + t}$, $t\geq 0$, considered on $\widetilde{\Omega}$, which is adapted to $\widetilde{\mathbb{F}} = (\mathcal{F}_t \cap \widetilde{\Omega})_{t \geq 0}$, is again a solution of Eq. (\[eqn:SDE\_mvd\]) but started at $x$. Hence, we can apply Theorem \[theorem:reflection\_and\_absorbing\] to obtain the statements of the corollary.

Concerning the question of non-existence of solutions of Eq. (\[eqn:SDE\_mvd\]), we see from Theorem \[theorem:reflection\_and\_absorbing\] and its Corollary \[cor:reflection\_and\_absorbing\] that the existence of a point $x \in \mathbb{R}$ with $\nu(\{x\}) > 1/2$ (resp., $\nu(\{x\}) < -1/2$, $|\nu(\{x\})| > 1$), in general, does not imply automatically that there is no solution. In general, there can be solutions which do not reach $x$ or, if $b(x)=0$ is satisfied, which stay in $x$ after reaching this level. But it holds the following

\[cor:non\_existence\_of\_a\_solution\] If $x \in \mathbb{R}$ is such that $\nu(\{x\}) > 1/2$ (resp., $\nu(\{x\}) < -1/2$, $|\nu(\{x\})| > 1$) and $b(x) \neq 0$, then there is no solution $(X,\mathbb{F})$ of Eq. (\[eqn:SDE\_mvd\]) with right (resp., left, symmetric) local time and arbitrary initial condition $X_0$ which satisfies $\mathbf{P}(\{\tau_x < +\infty\}) > 0$. In particular, there is no solution started at $X_0 = x$.

\(i) In the 