Heat kernels of non-symmetric Lévy-type operators

Abstract

We construct the fundamental solution (the heat kernel) $p^κ$ to the equation $\partial_t=\mathcal{L}^κ$, where under certain assumptions the operator $\mathcal{L}^κ$ 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 )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, $J\colon \mathbb{R}^d \to [0,\infty]$ is a Lévy density, i.e., $\int_{\mathbb{R}^d}(1\land |x|^2)J(x)dx<\infty$. The function $κ(x,z)$ is assumed to be Borel measurable on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<κ_0\leq κ(x,z)\leq κ_1$, and $|κ(x,z)-κ(y,z)|\leq κ_2|x-y|^β$ for some $β\in (0, 1)$. We prove the uniqueness, estimates, regularity and other qualitative properties of $p^κ$.