Fractional Laplacian with supercritical killings

Abstract

In this paper, we study Feynman-Kac semigroups of symmetric $\alpha$-stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form $b|x{|}^{-\beta }$, where $b>0$ and $\beta >\alpha$. We obtain two-sided estimates on the densities $p(t,x,y)$ of these semigroups for all $t>0$, along with estimates for the corresponding Green functions.

55 pages