Hölder regularity and gradient estimates 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 $(Z_t{)}_{t\ge 0}$ 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.

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References in corpus (2)

• Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts
• Schauder's estimate for nonlocal kinetic equations and its applications

Cited by in corpus (1)

• Schauder's estimates for nonlocal equations with singular Lévy measures