Heat kernels of non-symmetric Lévy-type operators

Abstract

We construct the fundamental solution (the heat kernel) $p^{\kappa }$ to the equation ${\partial }_t={\mathcal{L}}^{\kappa }$, where under certain assumptions the operator ${\mathcal{L}}^{\kappa }$ takes one of the following forms, \begin{align*} \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)κ(x,z)J(z)\, dz \,, \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x))κ(x,z)J(z)\, dz\,, \mathcal{L}^κf(x)&:= \frac1{2}\int_{\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))κ(x,z)J(z)\, dz\,. \end{align*} In particular, $J:{\mathbb{R}}^d\to [0,\infty ]$ is a Lévy density, i.e., ${\int }_{{\mathbb{R}}^d}(1\wedge |x{|}^2)J(x)dx<\infty$. The function $\kappa (x,z)$ is assumed to be Borel measurable on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ satisfying $0<{\kappa }_0\le \kappa (x,z)\le {\kappa }_1$, and $|\kappa (x,z)-\kappa (y,z)|\le {\kappa }_2|x-y{|}^{\beta }$ for some $\beta \in (0,1)$. We prove the uniqueness, estimates, regularity and other qualitative properties of $p^{\kappa }$.