The Laplace method for small deviations of Gaussian processes of Wiener type

V. R. Fatalov

Sbornik. Mathematics, Tome 196 (2005) no. 4, pp. 595-620

Résumé

Results on the exact asymptotics of the probabilities $$ \mathsf P\biggl\{\,\int_0^1|\xi(t)|^p\,dt \le\varepsilon^p\biggr\},\qquad\varepsilon\to 0, $$ for $p>0$ are proved for two Gaussian processes $\xi(t)$: the Wiener process and the Brownian bridge. The method of study is the Laplace method in Banach spaces and the approach to the probabilities of small deviations based on the theory of large deviations for the occupation time. The calculations are carried out for the cases $p=1$ and $p=2$ as a result of solving the extremal problem for the action functional and studying the corresponding Schrödinger equations.