Fundamental solutions of nonlocal H\"ormander's operators

Abstract

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$, $$ \cL^{(\alpha)}_{\sigma,b} f(x):=\mbox{p.v.} \int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\dif z+b(x)\cdot\nabla f(x)+\sL f(x), $$ where $\sigma:\mR^d\to\mR^d\times\mR^d$ and $b:\mR^d\to\mR^d$ are two $C^\infty_b$-functions, $\delta$ is a small positive number, p.v. stands for the Cauchy principal value, and $\sL$ 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 $j\in\mN$. Under the following uniform H\"ormander's type condition: for some $j_0\in\mN$, $$ \inf_{x\in\mR^d}\inf_{|u|=1}\sum_{j=1}^{j_0}|u B_j(x)|^2>0, $$ by using Bismut's approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator $\cL^{(\alpha)}_{\sigma,b}$. In particular, we answer a question proposed by Nualart \cite{Nu1} and Varadhan \cite{Va}.