Path integrals for fractional Brownian motion and fractional Gaussian noise.

Abstract

Wiener's path integral plays a central role in the study of Brownian motion. Here we derive exact path-integral representations for the more general fractional Brownian motion (FBM) and for its time derivative process, 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. Their exact path-integral representations were previously unknown. Our formalism exploits the Gaussianity of the FBM and FGN, relies on the 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 subdiffusive 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 make inroads in the study of the FBM and FGN.