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For a sufficiently regular Itô diffusion with $D=\sigma {\sigma }^T/2$, a density $\pi$ is stationary if and only if it solves the stationary Fokker-Planck equationwith integrable Fokker-Planck probability current and boundary conditions that make its outward flux vanish. Here $(\nabla \cdot (D\pi ){)}_i={\sum }_j{\partial }_j(D_{ij}\pi )$.

For the displayed parametrization, assumewhere $Z<\infty$, that $D(x)$ is symmetric positive semidefinite, and that $Q(x)$ is antisymmetric. Substituting $\nabla \pi =-\pi \nabla H$ and the stated $\Gamma$ into the probability current cancels all $D$ terms. The remaining divergence isbecause the second derivatives are symmetric in $i,j$ whereas $Q_{ij}=-Q_{ji}$. Thus these conditions, together with the boundary and regularity assumptions, imply stationarity. More generally, the divergence equation above is the exact necessary and sufficient condition; within this construction, $\pi \propto e^{-H}$ and antisymmetric $Q$ are the standard way to satisfy it.

The Fokker-Planck equation corresponds to the Itô diffusionabsorbed at $0$ and reflected at $L$.

For $v(x)=x$, an Euler--Maruyama proposal isIf $Y\le 0$, kill the path. If $X_n>0$ and $Y>0$, an endpoint-only test can miss a crossing. Conditional on the endpoints, the local Brownian bridge crossing probability isKill the path with this probability; otherwise impose reflection at $L$ by replacing an overshoot $Y>L$ with $2L-Y$ and set $X_{n+1}=Y$. Repeated reflection handles very rare multiple overshoots, and the approximation converges as $\Delta t\to 0$.