Maximum principles for time-fractional Cauchy problems with spatially non-local components

TL;DR

This paper establishes maximum principles and estimates for weak solutions of time-fractional PDEs with non-local spatial operators, introducing a new weak solution framework and probabilistic representations.

Contribution

It introduces a novel weak solution concept, proves existence and uniqueness, and links solutions to time-changed Brownian motion for fractional PDEs with non-local operators.

Findings

Proved a strong maximum principle for the problem.

Derived an Alexandrov-Bakelman-Pucci estimate.

Provided a probabilistic representation of solutions.

Abstract

We show a strong maximum principle and an Alexandrov-Bakelman-Pucci estimate for the weak solutions of a Cauchy problem featuring Caputo time-derivatives and non-local operators in space variables given in terms of Bernstein functions of the Laplacian. To achieve this, first we propose a suitable meaning of a weak solution, show their existence and uniqueness, and establish a probabilistic representation in terms of time-changed Brownian motion. As an application, we also discuss an inverse source problem.