The general Fokker-Planck equation associated with the SDE $dx_t = f(x_t) dt + \sigma(x_t)dW_t$ reads
$\partial_t \rho_t(x) = -\nabla\cdot(f\rho_t) + \frac{1}{2}\nabla\cdot\nabla\cdot(\sigma\sigma^T \rho_t).$
When $f = -\nabla H$ for some function $H$ and $\sigma\sigma^T = \beta^{-1} I$, we can show that under some conditions, the unique invariant distribution is the Gibbs distribution $\rho_t(x) \propto \exp(-\beta H)$. Is there a way to derive the stationary distribution for the more general case?