From non-Brownian Functionals to a Fractional Schr\"odinger Equation

TL;DR

This paper derives fractional Schr"odinger equations for anomalous diffusion functionals, connecting them to classical results and exploring applications like occupation times and ergodicity breaking.

Contribution

It introduces fractional Schr"odinger equations for path functionals of anomalous diffusion, extending the Feynman-Kac framework to non-Brownian processes.

Findings

Derived fractional Schr"odinger equations for anomalous diffusion functionals

Connected fractional equations to classical Feynman-Kac results in the normal diffusion limit

Analyzed occupation times and demonstrated relation to weak ergodicity breaking

Abstract

We derive backward and forward fractional Schr\"odinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.