= Solution
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 equation>
$$
\nabla\mathbin\cdot\left(b\pi-\nabla\mathbin\cdot(D\pi)\right)=0,
$$
with integrable <Fokker-Planck probability current> and boundary conditions that make its outward flux vanish. Here $(\nabla\mathbin\cdot(D\pi))_i=\sum_j\partial_j(D_{ij}\pi)$.
For the displayed parametrization, assume
$$
\pi(x)=Z^{-1}e^{-H(x)},
$$
where $Z<\infty$, 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 is
$$
-\sum_{i,j}\partial_i\partial_j(Q_{ij}\pi)=0,
$$
because 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.
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