We consider a Fokker-Planck equation: $$ \partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0, $$ with initial condition $m(0,\cdot) = \mu_0$, where $\mu_0$ is distribution with finite 2nd moment (but may not have a density).

Under general condition on $b$, we know that the Fokker-Planck equation has a unique weak solution (in sense of distribution). We wonder under which condition on $b$ we can expect to have a $C^{1,1,2}$ solution on $(0, \infty) \times \mathbb{R}^n \times \mathbb{R}^n$.

As we know, under the parabolic Hormander condition, we can obtain $C^\infty$ regularity. However, we prefer not to assume that $b$ is smooth in $t$ (we do assume $b$ is continuous in $t$). Do you know the literature providing the desired regularity?

Consider a Fokker-Planck equation: $$ \partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0, $$ with initial condition $m(0,\cdot,\cdot) = \mu_0$, where $\mu_0$ is a distribution with finite 2nd moment (but not necessarily a density with respect to the Lebesgue measure).

Under general condition on $b$, it is known that the Fokker-Planck equation has a unique weak solution (in the sense of distributions). I wonder under which condition on $b$ one can expect to have a $C^{1,1,2}$ solution on $(0, \infty) \times \mathbb{R}^n \times \mathbb{R}^n$?

Under the parabolic Hormander condition one can classically obtain $C^\infty$ regularity. However, I prefer not to assume that $b$ is smooth in $t$ (possibly discontinuous in time). Does anyone know of literature providing such regularity?