On the space-like analyticity in the extension problem for nonlocal parabolic equations

TL;DR

This paper proves that solutions to certain fractional parabolic equations are real-analytic in space variables, especially in the extension variable, and uses this to establish unique continuation properties for these nonlocal operators.

Contribution

It provides an elementary proof of space-like real analyticity for solutions to fractional parabolic equations and applies this to prove weak unique continuation.

Findings

Solutions are real-analytic in the extension variable and all space variables.

Weak solutions that are even in the extension variable are real-analytic in space.

The weak unique continuation property holds for nonlocal parabolic operators with analytic coefficients.

Abstract

In this note we give an elementary proof of the space-like real analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type $(\partial_t - div_x(B(x)\nabla_x))^s$, $0<s<1$. Our primary interest is in the so-called \emph{extension variable}. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where $B(x)$ is a uniformly elliptic matrix-valued function with real-analytic entries.