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

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: \[\mathscr{L}^{(α)}_b:=Δ^{α/2}_{\mathrm{v}}+\mathrm {v}\cdot \nabla_x+b(x,\mathrm{v})\cdot \nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^d,\] 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.

Published in at http://dx.doi.org/10.1214/13-AOP900 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (4)

• Quantifying model uncertainty for the observed non-Gaussian data by the Hellinger distance
• Nonlocal Hormander's hypoellipticity theorem
• Regularity of density for SDEs driven by degenerate Lévy noises
• Fundamental solutions of nonlocal Hörmander's operators