Path integrals for fractional Brownian motion and fractional Gaussian noise

TL;DR

This paper derives exact path-integral representations for fractional Brownian motion and fractional Gaussian noise, enabling new analytical approaches to study these non-Markovian stochastic processes with wide-ranging applications.

Contribution

It provides the first explicit path-integral formulations for fBm and fGn, extending the Wiener integral framework to these complex processes.

Findings

Exact path-integral representations for fBm and fGn derived.

Formalism incorporates external forcing effects.

Facilitates advanced analysis of anomalous diffusion and related phenomena.

Abstract

The Wiener's path integral plays a central role in the studies of Brownian motion. Here we derive exact path-integral representations for the more general \emph{fractional} Brownian motion (fBm) and for its time derivative process -- the fractional Gaussian noise (fGn). These paradigmatic non-Markovian stochastic processes, introduced by Kolmogorov, Mandelbrot and van Ness, found numerous applications across the disciplines, ranging from anomalous diffusion in cellular environments to mathematical finance. Still, their exact path-integral representations were previously unknown. Our formalism exploits the Gaussianity of the fBm and fGn, relies on theory of singular integral equations and overcomes some technical difficulties by representing the action functional for the fBm in terms of the fGn for the sub-diffusive fBm, and in terms of the derivative of the fGn for the super-diffusive fBm. We also extend the formalism to include external forcing. The exact and explicit path-integral representations open new inroads into the studies of the fBm and fGn.