Fundamental solution for super-critical non-symmetric L\'evy-type operators

TL;DR

This paper establishes the existence and estimates of the fundamental solution for a class of non-symmetric, non-local operators with stable-like behavior, especially when the operator's order is less than or equal to one, under broad conditions.

Contribution

It introduces a novel approach to handle non-symmetry in non-local operators by imposing cancellation conditions, extending results even to the 1-stable Lévy measure case.

Findings

Proved existence of the heat kernel for broad classes of operators.

Provided estimates for the fundamental solution under minimal assumptions.

Extended results to the case of 1-stable Lévy measures.

Abstract

We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation $\partial_t =\mathcal{L}^{\kappa}$ for non-symmetric non-local operators $$ \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz\,, $$ under broad assumptions on $\kappa$ and $J$. Of special interest is the case when the order of the operator $\mathcal{L}^{\kappa}$ is smaller than or equal to 1. Our approach rests on imposing suitable cancellation conditions on the internal drift coefficient $$ \int_{r\leq |z|<1} z \kappa(x,z)J(z)dz\,,\qquad 0<r\leq 1\,, $$ which allows us to handle the non-symmetry of $z\mapsto \kappa(x,z)J(z)$. The results are new even for the $1$-stable L\'evy measure $J(z)=|z|^{-d-1}$.