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 ${\partial }_t={\mathcal{L}}^{\kappa }$ for non-symmetric non-local operators $${\mathcal{L}}^{\kappa }f(x):={\int }_{{\mathbb{R}}^d}(f(x+z)-f(x)-1_{|z|<1}⟨z,\nabla f(x)⟩)\kappa (x,z)J(z)dz,$$ 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 $${\int }_{r\le |z|<1}z\kappa (x,z)J(z)dz,0<r\le 1,$$ 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 Lévy measure $J(z)=|z{|}^{-d-1}$.

arXiv commentarXiv admin note: text overlap with arXiv:1804.01313