Path integral representation for fractional Brownian motion

Abstract

Fractional Brownian motion (FBM) is a generalization of the usual Brownian motion. A path integral representation that has recently been suggested for it is shown to be not for the FBM but for a different generalization of the Brownian motion. A new path integral representation is given and its measure has fractional derivatives of the path in it. The measure shows that the process is Gaussian but is, in general, non-Markovian, even though Brownian motion itself is Markovian. It is shown how the propagator for the motion of free FBM may be evaluated. This is somewhat more complex than for the usual path integrals, due to the occurrence of fractional derivatives. We also find the propagator in the presence of a linear absorption (potential), and for FBM on a ring.