The Harnack inequality for a class of nonlocal parabolic equations

TL;DR

This paper proves a scale-invariant Harnack inequality for fractional powers of certain nonlocal parabolic operators, extending classical results to non-Euclidean contexts and providing new tools for analysis of such equations.

Contribution

It introduces a novel scale-invariant Harnack inequality for fractional nonlocal parabolic operators in non-Euclidean settings, expanding the theoretical framework.

Findings

Established Harnack inequality for $( ext{partial}_t - ext{L})^s$

Derived similar results for $(- ext{L})^s$

Applicable to non-Euclidean geometries

Abstract

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