Geometrical optics of large deviations of fractional Brownian motion.

Abstract

It has been shown recently that the optimal fluctuation method-essentially geometrical optics-provides a valuable insight into large deviations of Brownian motion. Here we extend the geometrical optics formalism to two-sided, -∞<t<∞, fractional Brownian motion (fBm) on the line, which is "pushed" to a large deviation regime by imposed constraints. We test the formalism on three examples where exact solutions are available: the two- and three-point probability distributions of the fBm and the distribution of the area under the fBm on a specified time interval. Then we apply the formalism to several previously unsolved problems by evaluating large-deviation tails of the following distributions: (i) of the first-passage time, (ii) of the maximum of, and (iii) of the area under, fractional Brownian bridge and fractional Brownian excursion, and (iv) of the first-passage area distribution of the fBm. An intrinsic part of a geometrical optics calculation is determination of the optimal path-the most likely realization of the process which dominates the probability distribution of the conditioned process. Due to the non-Markovian nature of the fBm, the optimal paths of a fBm, subject to constraints on a finite interval 0<t≤T, involve both the past -∞<t<0 and the future T<t<∞.