Equivalence of definitions of fractional caloric functions

TL;DR

This paper establishes the equivalence between different definitions of fractional caloric functions, providing conditions for classical solutions and estimates for heat kernel derivatives.

Contribution

It proves the equivalence of distributional and mean value solutions for the fractional heat equation and offers new estimates for heat kernel derivatives.

Findings

Proves equivalence between distributional and mean value solutions.

Provides conditions for classical solutions with boundary data.

Derives off-diagonal estimates for heat kernel derivatives.

Abstract

We prove equivalence between nonnegative distributional solutions of the fractional heat equation and caloric functions, i.e., functions satisfying the mean value property with respect to the space-time isotropic $\alpha$-stable process. We also provide sufficient conditions for the boundary and exterior data under which the solutions are classical and we give off-diagonal estimates for the derivatives of the Dirichlet heat kernel and the lateral Poisson kernel, which might be of their own interest.