The Annals of Probability

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

Xicheng Zhang

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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.

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

First available in Project Euclid: 29 August 2014

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Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 35Q84: Fokker-Planck equations

Malliavin calculus Hörmander’s condition $\alpha$-stable process distributional density SDE


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

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