Nonnegative solutions to nonlocal parabolic equations
Analysis of PDEs
2025-05-14 v1 Probability
Abstract
We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a result has previously been studied only for certain translation invariant operators, and new ideas are needed in our general setting. Second, we establish sharp two-sided bounds for the fundamental solution via purely variational techniques, entirely bypassing tools from semigroup theory, Dirichlet forms, and stochastic analysis. Third, we derive sharp Harnack-type estimates that are novel even for the fractional heat equation.
Cite
@article{arxiv.2505.08449,
title = {Nonnegative solutions to nonlocal parabolic equations},
author = {Naian Liao and Marvin Weidner},
journal= {arXiv preprint arXiv:2505.08449},
year = {2025}
}