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The Wiener's path integral plays a central role in the studies of Brownianmotion. 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-Markovianstochastic processes, introduced by Kolmogorov, Mandelbrot and van Ness, foundnumerous applications across the disciplines, ranging from anomalous diffusionin cellular environments to mathematical finance. Still, their exactpath-integral representations were previously unknown. Our formalism exploitsthe Gaussianity of the fBm and fGn, relies on theory of singular integralequations and overcomes some technical difficulties by representing the actionfunctional for the fBm in terms of the fGn for the sub-diffusive fBm, and interms of the derivative of the fGn for the super-diffusive fBm. We also extendthe formalism to include external forcing. The exact and explicit path-integralrepresentations open new inroads into the studies of the fBm and fGn.

Path integrals for fractional Brownian motion and fractional Gaussian noise