Given a Markov process $(X_t)_{t\geq 0}$ on $(\mathbb R^n, \mathcal B_{\mathbb R^n})$, under which conditions does the solution to the Fokker-Planck equation
$$\frac{\partial u(t,x)}{\partial t}= \mathcal L^* u(t,x)$$
admit a density with respect to the Lebesgue measure?
I am particularly interested in the case where $X_t$ is a diffusion process, but I would prefer a general answer if that exists. In the diffusion case, the condition might be something along ellipticity or uniform ellipticity. A reference would be much appreciated. Thanks.