The Harnack inequality for a class of nonlocal parabolic equations

Abstract

In this paper we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators $({\partial }_t-\mathcal{L}{)}^s$, $0<s<1$, where $\mathcal{L}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product we also obtain a similar result for the nonlocal operators $(-\mathcal{L}{)}^s$. Our focus is on non-Euclidean situations.