## The Annals of Probability

### Densities for SDEs driven by degenerate $\alpha$-stable processes

Xicheng Zhang

#### 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.

#### Article information

Source
Ann. Probab., Volume 42, Number 5 (2014), 1885-1910.

Dates
First available in Project Euclid: 29 August 2014

https://projecteuclid.org/euclid.aop/1409319469

Digital Object Identifier
doi:10.1214/13-AOP900

Mathematical Reviews number (MathSciNet)
MR3262494

Zentralblatt MATH identifier
1307.60090

#### Citation

Zhang, Xicheng. Densities for SDEs driven by degenerate $\alpha$-stable processes. Ann. Probab. 42 (2014), no. 5, 1885--1910. doi:10.1214/13-AOP900. https://projecteuclid.org/euclid.aop/1409319469

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