Nonnegative solutions to nonlocal parabolic equations

TL;DR

This work develops a complete nonlocal parabolic theory with bounded measurable, time-dependent kernels: it constructs a fundamental solution, proves two-sided polynomial bounds via variational methods, and establishes a Widder-type theorem giving a unique initial trace for nonnegative global solutions. It then derives sharp Harnack-type estimates for global solutions, including a novel off-diagonal control that remains valid for time-dependent kernels. The results hold under mild coercivity assumptions, do not rely on semigroup or stochastic tools, and extend known results beyond translation-invariant or locally regular kernels. Altogether, the paper provides a robust, variational framework for the structure, growth, and regularity of nonlocal diffusion processes described by $\partial_t u - \mathcal{L}_t u = 0$.

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.

Theorems & Definitions (49)

• theorem 1: Widder-type theorem
• remark 2
• theorem 3
• theorem 4: Harnack-type estimate
• remark 5
• remark 6
• remark 7
• theorem 8: Growth estimate via initial data
• definition 9: local solution
• definition 10: global solution