Fundamental solutions of nonlocal Hörmander's operators. II.
type: Item
instance of: scholarly article
title: Fundamental solutions of nonlocal Hörmander's operators. II.
zbMATH Open document ID: 1372.60098
DOI: 10.1214/16-AOP1102
author: Xicheng Zhang
published in: The Annals of Probability
publication date: 2017-07-28T00:00:00Z
review text: The author considers nonlocal integro-differential Hörmander operator with smooth coefficients and introduces uniform Hörmander-type conditions on the coefficients of this operator. Under these conditions, using \textit{J.-M. Bismut}'s [Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 147--235 (1983; Zbl 0494.60082)] approach to the Malliavin calculus with jumps, the existence of fundamental solutions to the Hörmander operator is proved. From the point of view of stochastic differential equations (SDEs), the problem is equivalent to the study of distributional density of the solution of SDE with \(d\)-dimensional Brownian motion and independent stable subordinator. The smoothness of the distributional density is established for nonconstant diffusion coefficient. For Part I, see [\textit{X. Zhang}, Commun. Math. Stat. 4, No. 3, 359--402 (2016; Zbl 1365.60064)].
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 47G20
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 6754787
zbMATH Keywords: nonlocal integro-differential Hörmander operator
zbMATH Keywords: smooth coefficients
zbMATH Keywords: Hörmander condition
zbMATH Keywords: fundamental solution
zbMATH Keywords: Malliavin calculus with jumps
zbMATH Keywords: Malliavin matrix
zbMATH Keywords: stable subordinator
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6754787