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

V. R. Fatalov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 4, pp. 725-772

Résumé

This paper presents a survey of results on computing the small deviation asymptotics for Gaussian measures, that is, the asymptotics of the probabilities $$ \mu(\varepsilon D), \qquad \varepsilon\to0, $$ where $D$ is a bounded domain in a Banach space $(B,{\|\cdot\|})$ (for example, $D=\{x\in B:\|x\|\leqslant 1\}$) and $\mu$ a Gaussian measure on $B$. 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, $n$-parameter fractional Brownian motion, $n$-parameter Wiener–Chentsov fields, and the Wiener pillow. Results on small deviations are presented in diverse norms, namely, the sup-norm, Hilbert norms, $L^p$-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.