Nonnegative solutions to nonlocal parabolic equations

2025/05/13 by Liao, Naian, Weidner, Marvin · 1 citation
#31B05 #35A02 #35B65 #35C15 #35K08 #35R09 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2505.08449

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.

Citations

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Cited by

• Liouville theorem for singular solutions to nonlocal equations

Related

• The parabolic Harnack inequality for nonlocal equations
• The parabolic Harnack inequality for nonlocal equations
• On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions
• Harnack's Inequality for Parabolic Nonlocal Equations
• Dirichlet heat kernel estimates for parabolic nonlocal equations