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

TL;DR

This paper proves the existence and smoothness of probability density functions for solutions to certain degenerate stochastic differential equations driven by alpha-stable processes, using Malliavin calculus under Hörmander's condition.

Contribution

It establishes the existence of densities and smooth heat kernels for SDEs driven by degenerate subordinated Brownian motions, extending to fractional kinetic Fokker-Planck operators.

Findings

Existence of distributional densities under Hörmander's condition.

Smoothness of densities in a special degenerate case.

Existence of smooth heat kernels for fractional kinetic operators.

Abstract

In this work, by using the Malliavin calculus, under H\"ormander'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.