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Nonnegative solutions to nonlocal parabolic equations

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arxiv 2505.08449 v1 pith:BHMZ7Q5O submitted 2025-05-13 math.AP math.PR

Naian Liao , Marvin Weidner This is my paper · ORCID
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keywords equationsgeneralnonlocalnonnegativeparabolicsharpsolutionsanalysis
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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.

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  1. Liouville theorem for singular solutions to nonlocal equations

    math.AP 2025-07 conditional novelty 7.0 of 10

    Every singular solution to a nonlocal linear equation with measurable kernel that is one-sided bounded near zero and infinity must equal a multiple of the fundamental solution plus a constant.

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