The Annals of Probability

Fundamental solutions of nonlocal Hörmander’s operators II

Xicheng Zhang

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
Revised: January 2016
First available in Project Euclid: 15 May 2017

https://projecteuclid.org/euclid.aop/1494835232

Digital Object Identifier
doi:10.1214/16-AOP1102

Mathematical Reviews number (MathSciNet)
MR3650416

Zentralblatt MATH identifier
1372.60098

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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