A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift

Abstract: Despite there are numerous theoretical studies of stochastic differential equations with a symmetric $\alpha$-stable L\'evy noise, very few regularity results exist in the case of $0<\alpha\leq1$. In this paper, we study the fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift, and prove that there exists a unique solution, which is $C^\infty$ in space for $t>0$ when $\alpha\in (0, 2]$.