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.
Nonnegative solutions to nonlocal parabolic equations
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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.
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Liouville theorem for singular solutions to nonlocal equations
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.