Nonlocal Hormander's hypoellipticity theorem

Abstract

Consider the following nonlocal integro-differential operator: for $\alpha \in (0,2)$, $$ \cal L^{(α)}_{σ,b} f(x):=\mbox{p.v.} \int_{\mathbb{R}^d-\{0\}}\frac{f(x+σ(x)z)-f(x)}{|z|^{d+α}}d z+b(x)\cdot\nabla f(x), $$ where $\sigma :{\mathbb{R}}^d\to {\mathbb{R}}^d\times {\mathbb{R}}^d$ and $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ are two $C_b^{\infty }$-functions, 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 $n=n(x)\in \mathbb{N}$, $$\mathrm{Rank}[B_1(x),B_2(x),\cdots ,B_n(x)]=d,$$ by using the Malliavin calculus, we prove the existence of the heat kernel ${\rho }_t(x,y)$ to the operator ${\mathcal{L}}_{\sigma ,b}^{(\alpha )}$ as well as the continuity of $x\mapsto {\rho }_t(x,\cdot )$ in $L^1({\mathbb{R}}^d)$ for each $t>0$. Moreover, when $\sigma (x)=\sigma$ is constant, under the following uniform Hörmander's type condition: for some $j_0\in \mathbb{N}$, $$\underset{x\in {\mathbb{R}}^d}{\inf }\underset{|u|=1}{\inf }\sum _{j=1}^{j_0}|u{B}_j(x){|}^2>0,$$ we also prove the smoothness of $(t,x,y)\mapsto {\rho }_t(x,y)$ with ${\rho }_t(\cdot ,\cdot )\in C_b^{\infty }({\mathbb{R}}^d\times {\mathbb{R}}^d)$ for each $t>0$.

30pp. Correct some typos

References in corpus (6)

• Densities for SDEs driven by degenerate $\alpha$-stable processes
• Derivative formula and gradient estimate for SDEs driven by $\alpha$-stable processes
• Harnack Inequalities for Stochastic Equations Driven by Lévy Noise
• On the distance between probability density functions
• Strong Feller properties for degenerate SDEs with jumps
• Fundamental solution of kinetic Fokker-Planck operator with anisotropic nonlocal dissipativity

Cited by in corpus (3)

• Regularity of density for SDEs driven by degenerate Lévy noises
• Fundamental solutions of nonlocal Hörmander's operators
• Smooth densities of stochastic differential equations forced by degenerate stable type noises