The Annals of Probability

Fundamental solutions of nonlocal Hörmander’s operators II

Xicheng Zhang

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Abstract

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$: \[\mathcal{L}^{(\alpha)}_{\sigma,b}f(x):=\mbox{p.v.}\int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\,\mathrm{d}z+b(x)\cdot\nabla f(x)+{\mathscr{L}}f(x),\] where $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ and $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are smooth functions and have bounded partial derivatives of all orders greater than $1$, $\delta$ is a small positive number, p.v. stands for the Cauchy principal value and ${\mathscr{L}}$ is a bounded linear operator in Sobolev spaces. Let $B_{1}(x):=\sigma(x)$ and $B_{j+1}(x):=b(x)\cdot\nabla{B}_{j}(x)-\nabla{b(x)}\cdot B_{j}(x)$ for $j\in\mathbb{N}$. Suppose $B_{j}\in C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ for each $j\in\mathbb{N}$. Under the following uniform Hörmander’s type condition: for some $j_{0}\in\mathbb{N}$, \[\inf_{x\in\mathbb{R}^{d}}\inf_{|u|=1}\sum_{j=1}^{j_{0}}|uB_{j}(x)|^{2}>0,\] by using Bismut’s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator $\mathcal{L}^{(\alpha)}_{\sigma,b}$. In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46–49] and Varadhan [Sankhyā A 73 (2011) 50–51].

Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1799-1841.

Dates
Received: October 2014
Revised: January 2016
First available in Project Euclid: 15 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1494835232

Digital Object Identifier
doi:10.1214/16-AOP1102

Mathematical Reviews number (MathSciNet)
MR3650416

Zentralblatt MATH identifier
1372.60098

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Hörmander condition fundamental solution Malliavin calculus with jumps nonlocal kinetic Fokker–Planck operator Poisson functional

Citation

Zhang, Xicheng. Fundamental solutions of nonlocal Hörmander’s operators II. Ann. Probab. 45 (2017), no. 3, 1799--1841. doi:10.1214/16-AOP1102. https://projecteuclid.org/euclid.aop/1494835232


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