On distributions of functionals of anomalous diffusion paths

TL;DR

This paper derives a fractional Feynman-Kac equation for functionals of anomalous diffusion paths, extending classical results to non-Brownian motion and providing solutions for various applications.

Contribution

It introduces a fractional generalization of the Feynman-Kac equation for anomalous diffusion, including derivations of backward equations and extensions to Levy flights.

Findings

Derived fractional Feynman-Kac equation for anomalous paths

Provided solutions for occupation time, first passage time, and maximal displacement

Extended the framework to Levy flights and discussed related fractional Schrödinger equations

Abstract

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Levy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrodinger equations that recently appeared in the literature.