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Fundamental Solutions of Nonlocal Hörmander’s Operators

Research paper by Xicheng Zhang

Indexed on: 16 Sep '16Published on: 07 Sep '16Published in: Communications in Mathematics and Statistics



Abstract

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 \(t>0\) .AbstractConsider 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 \(t>0\) . \(\alpha \in (0,2)\) \(\alpha \in (0,2)\) $$\begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{ z ^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned}$$ $$\begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{ z ^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned}$$ \(\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d\) \(\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d\) \(b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\) \(b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\) \(B_1(x):=\sigma (x)\) \(B_1(x):=\sigma (x)\) \(B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)\) \(B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)\) \(j\in {\mathbb {N}}\) \(j\in {\mathbb {N}}\) \(x\in {\mathbb {R}}^d\) \(x\in {\mathbb {R}}^d\) \(n=n(x)\in {\mathbb {N}}\) \(n=n(x)\in {\mathbb {N}}\) $$\begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned}$$ $$\begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned}$$ \(\rho _t(x,y)\) \(\rho _t(x,y)\) \({\mathcal {L}}^{(\alpha )}_{\sigma ,b}\) \({\mathcal {L}}^{(\alpha )}_{\sigma ,b}\) \(x\mapsto \rho _t(x,\cdot )\) \(x\mapsto \rho _t(x,\cdot )\) \(L^1({\mathbb {R}}^d)\) \(L^1({\mathbb {R}}^d)\) \(t>0\) \(t>0\) \(\sigma (x)=\sigma \) \(\sigma (x)=\sigma \) \(B_j\in C^\infty _b\) \(B_j\in C^\infty _b\) \(j\in {\mathbb {N}}\) \(j\in {\mathbb {N}}\) \(j_0\in {\mathbb {N}}\) \(j_0\in {\mathbb {N}}\) $$\begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{ u =1}\sum _{j=1}^{j_0} u B_j(x) ^2>0, \end{aligned}$$ $$\begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{ u =1}\sum _{j=1}^{j_0} u B_j(x) ^2>0, \end{aligned}$$ \((t,x,y)\mapsto \rho _t(x,y)\) \((t,x,y)\mapsto \rho _t(x,y)\) \(\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)\) \(\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)\) \(t>0\) \(t>0\)