Heat kernels of non-symmetric L\'evy-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}^{\kappa}f(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz \,, \mathcal{L}^{\kappa}f(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x))\kappa(x,z)J(z)\, dz\,, \mathcal{L}^{\kappa}f(x)&:= \frac1{2}\int_{\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))\kappa(x,z)J(z)\, dz\,. \end{align*} In particular, $J\colon \mathbb{R}^d \to [0,\infty]$ is a L\'evy density, i.e., $\int_{\mathbb{R}^d}(1\land |x|^2)J(x)dx<\infty$. The function $\kappa(x,z)$ is assumed to be Borel measurable on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<\kappa_0\leq \kappa(x,z)\leq \kappa_1$, and $|\kappa(x,z)-\kappa(y,z)|\leq \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1)$. We prove the uniqueness, estimates, regularity and other qualitative properties of $p^{\kappa}$.