Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha $-stable processes

Abstract

We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: \[ \mathrm{d} X_{t}=\sigma (t, X_{t-})\mathrm{d} Z_{t}+b (t, X_{t})\mathrm{d} t,\ \ X_{0}=x\in{\mathbb {R}} ^{d}, \] where $( Z_{t})_{t\geqslant 0}$ is a $d$-dimensional cylindrical $\alpha $-stable process with $\alpha \in (0, 2)$, $\sigma (t, x):{\mathbb{R} }_{+}\times{\mathbb {R}} ^{d}\to{\mathbb {R}} ^{d}\otimes{\mathbb {R}} ^{d}$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb{R} }_{+}\times{\mathbb {R}} ^{d}\to{\mathbb {R}} ^{d}$ is bounded $\beta $-Hölder continuous in $x$ uniformly in $t$ with $\beta \in [0,1]$ satisfying $\alpha +\beta >1$. Moreover, we also show the existence and regularity of the distributional density of $X (t, x)$. Our proof is based on Littlewood-Paley’s theory.