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September 2014 Densities for SDEs driven by degenerate $\alpha$-stable processes
Xicheng Zhang
Ann. Probab. 42(5): 1885-1910 (September 2014). DOI: 10.1214/13-AOP900

Abstract

In this work, by using the Malliavin calculus, under Hörmander’s condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker–Planck (nonlocal) operator: \[\mathscr{L}^{(\alpha)}_{b}:=\Delta^{\alpha/2}_{\mathrm{v}}+\mathrm{v} \cdot \nabla_{x}+b(x,\mathrm{v})\cdot\nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^{d},\] where $\alpha\in(0,2)$ and $b:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is smooth and has bounded derivatives of all orders.

Citation

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Xicheng Zhang. "Densities for SDEs driven by degenerate $\alpha$-stable processes." Ann. Probab. 42 (5) 1885 - 1910, September 2014. https://doi.org/10.1214/13-AOP900

Information

Published: September 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1307.60090
MathSciNet: MR3262494
Digital Object Identifier: 10.1214/13-AOP900

Subjects:
Primary: 60H07 , 60H10
Secondary: 35Q84

Keywords: $\alpha$-Stable process , distributional density , Hörmander’s condition , Malliavin calculus , SDE

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 5 • September 2014
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