Given a family of densities $\{p(x, t)\}_t, t \in [0, \infty),$ with (tractable) stationary density $\lim_{t \to \infty} p(x,t) = p_\infty(x)$ under what conditions is the family a solution to the Fokker-Planck equation? That is, under what conditions do there exist drift $\mu(x, t)$ and diffusion $\sigma(x,t)$ such that
$$\partial_t p(x,t) = -\sum_{i=1}^d \partial_{x_i} [\mu_i(x,t) p(x,t)] + \frac{1}{2} \sum_{i=1}^d\sum_{j=1}^d \partial_{x_i} \partial_{x_j}[\sigma_{ik}(x, t) \sigma_{jk}(x,t) p(x,t)],$$
for all $t \in [0, \infty)$.
