Asymptotics of large deviations of Gaussian processes of Wiener type for -functionals, , and the hypergeometric function

Abstract

A general result is obtained on exact asymptotics of the probabilities as and for Gaussian processes . The general theorem is applied for the calculation of these asymptotics in the cases of the following processes: the Wiener process , the Brownian bridge, and the stationary Gaussian process , . The Laplace method in Banach spaces is used. The calculations of the constants reduce to solving an extremum problem for the action functional and studying the spectrum of a differential operator of the second order of Sturm-Liouville type.