Fractional Cauchy problems on bounded domains

TL;DR

This paper develops classical and stochastic solutions for fractional Cauchy problems with Dirichlet boundary conditions in bounded domains, linking fractional derivatives to inverse stable subordinators and iterated Brownian motion.

Contribution

It introduces a method to solve fractional Cauchy problems on bounded domains using inverse stable subordinators and establishes a connection with iterated Brownian motion.

Findings

Constructed stochastic solutions via inverse stable subordinators.

Established a correspondence between fractional derivatives and iterated Brownian motion.

Solved Dirichlet problems for fractional derivatives in bounded domains.

Abstract

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain $D\subset\mathbb{R}^d$ with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time.