Densities for sdes driven by degenerate α-stable processes

Xicheng Zhang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)

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: ℒ(α)b:= δα/2v + v · ∇x + b (x,v)· ∇v, x, v ε ℝd, where α ε (0, 2) and b:ℝd × ℝd →ℝd is smooth and has bounded derivatives of all orders.

Original languageEnglish
Pages (from-to)1885-1910
Number of pages26
JournalAnnals of Probability
Volume42
Issue number5
DOIs
Publication statusPublished - Sept 2014
Externally publishedYes

Keywords

  • Distributional density
  • Hörmander's condition
  • Malliavin calculus
  • SDE
  • α-Stable process

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