Fundamental solution for super-critical non-symmetric Lévy-type operators

Abstract

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