Path integral formulation of fractional Brownian motion for general Hurst exponent

TL;DR

This paper develops a unified path integral formulation for fractional Brownian motion applicable to all Hurst exponents, simplifying the derivation of propagators for both subdiffusive and superdiffusive cases.

Contribution

It introduces a novel, unified method to construct the path integral representation of fractional Brownian motion for any Hurst exponent, extending previous approaches.

Findings

Unified path integral formulation for all H values.

Simplified derivation of the propagator for fractional Brownian motion.

Applicable to both subdiffusive and superdiffusive regimes.

Abstract

In J. Phys. A: Math. Gen. 28, 4305 (1995), K. L. Sebastian gave a path integral computation of the propagator of subdiffusive fractional Brownian motion (fBm), i.e. fBm with a Hurst or self-similarity exponent $H\in(0,1/2)$. The extension of Sebastian's calculation to superdiffusion, $H\in(1/2,1]$, becomes however quite involved due to the appearance of additional boundary conditions on fractional derivatives of the path. In this paper, we address the construction of the path integral representation in a different fashion, which allows to treat both subdiffusion and superdiffusion on an equal footing. The derivation of the propagator of fBm for general Hurst exponent is then performed in a neat and unified way.