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2020 Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha $-stable processes
Zhen-Qing Chen, Zimo Hao, Xicheng Zhang
Electron. J. Probab. 25: 1-23 (2020). DOI: 10.1214/20-EJP542

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.

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Zhen-Qing Chen. Zimo Hao. Xicheng Zhang. "Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha $-stable processes." Electron. J. Probab. 25 1 - 23, 2020. https://doi.org/10.1214/20-EJP542

Information

Received: 20 March 2020; Accepted: 22 October 2020; Published: 2020
First available in Project Euclid: 18 November 2020

Digital Object Identifier: 10.1214/20-EJP542

Subjects:
Primary: 60G52, 60H10

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