Fundamental Solutions of Nonlocal Hörmander’s Operators

Abstract

Consider the following nonlocal integro-differential operator: for $\alpha \in (0,2)$,

$$\begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{|z|^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned}$$

where $\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d$ and $b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$ are smooth and have bounded first-order derivatives, and p.v. stands for the Cauchy principal value. 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 $j\in {\mathbb {N}}$. Under the following Hörmander’s type condition: for any $x\in {\mathbb {R}}^d$ and some $n=n(x)\in {\mathbb {N}}$,

$$\begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned}$$

by using the Malliavin calculus, we prove the existence of the heat kernel $\rho _t(x,y)$ to the operator ${\mathcal {L}}^{(\alpha )}_{\sigma ,b}$ as well as the continuity of $x\mapsto \rho _t(x,\cdot )$ in $L^1({\mathbb {R}}^d)$ as a density function for each $t>0$. Moreover, when $\sigma (x)=\sigma$ is constant and $B_j\in C^\infty _b$ for each $j\in {\mathbb {N}}$, under the following uniform Hörmander’s type condition: for some $j_0\in {\mathbb {N}}$,

$$\begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{|u|=1}\sum _{j=1}^{j_0}|u B_j(x)|^2>0, \end{aligned}$$

we also show the smoothness of $(t,x,y)\mapsto \rho _t(x,y)$ with $\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)$ for each $t>0$.