Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields

Abstract

This paper presents a survey of results on computing the small deviation asymptotics for Gaussian measures, that is, the asymptotics of the probabilities where is a bounded domain in a Banach space (for example, ) and a Gaussian measure on .The main attention is focused on calculating the values of constants in the exact or logarithmic asymptotics. The survey contains new numerical results; some erroneous assertions in previous papers on this topic are also noted.The following classes of Gaussian processes and fields are studied in detail: Wiener processes and related processes, Brownian bridges, Bessel processes, vector Wiener processes, Gaussian Markov processes, Gaussian processes with stationary increments, fractional Ornstein-Uhlenbeck processes, -parameter fractional Brownian motion, -parameter Wiener-Chentsov fields, and the Wiener pillow. Results on small deviations are presented in diverse norms, namely, the sup-norm, Hilbert norms, -norms, Hölder norms, Orlicz norms, and weighted sup-norms. About 30 problems concerned with finding exact constants in asymptotic expressions for small deviations are posed. The relation to Chung's law of the iterated logarithm is also considered, and a number of other results are presented.