Detailed balance for the Fokker-Planck equation

Question

Consider the Fokker-Planck equation $$\partial_{t} p=-\nabla \cdot(b(x) p)+\Delta p, x \in \mathbb{R}^{d}$$ where $\nabla\times b\neq0.$ If $\pi$ is the stationary solution, then we have $$\nabla\cdot(b(x)\pi+\nabla\pi)=0.$$It is known that the stationary solution satisfies $\pi > 0$ for all $x \in \mathbb{R}^d$ if $b$ is confining. Will the stationary solution satisfy $−b(x)\pi + \nabla \pi = 0$, i.e the detailed balance satisfy?

Answer 1

No. To simplify things, we suppose that $\pi$ has a density which we will denote also as $\pi$. The case of general $\pi$ is largely the same but a bit more convoluted as we would have to use distributions.

We suppose for contradiction that $\nabla \times b \neq 0$ and that $\pi$ satisfies $-b(x) \pi + \nabla \pi = 0$. Rearranging this equation we get $$-b(x) = \frac{\nabla \pi}{\pi}. $$ The right-hand side is the gradient of $\log{\pi}$ so it must have zero curl but the left-hand side has non-zero curl by assumption.