Asymptotics of large deviations for Wiener random fields in $L^p$-norm, nonlinear Hammerstein equations, and high-order hyperbolic boundary-value problems

V. R. Fatalov

Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 4, pp. 710-726

Résumé

This paper provides derivation, for $1<p\le 2$, of exact asymptotics as $u\to\infty$ of the probabilities $$ \mathbf{P}\biggl\{\biggl(\int_{[0,1]^n}|X(t)|^p\,dt\biggr)^{1/p}>u\biggr\} $$ for two Gaussin fields, namely, the Wiener field of Jech–Chentsov and the so-called “Brownian cushion,” being, respectively, the multiparameter analogues of the Wiener process and the Brownian bridge. These Gaussian fields have zero means, and their respective covariance functions are $\prod_{i=1}^n\min(t_i, s_i)$ and $\prod_{i=1}^n[\min(t_i,s_i)-t_is_i]$, $t=(t_1,\dots,t_n)$, $s=(s_1,\dots,s_n)$. The method of analysis is the Laplace method in Banach spaces. We display the relation of the problem under consideration with the theory of nonlinear Hammerstein equations in $\mathbf{R}^n $ and the hyperbolic boundary-value problems of high order. Solutions of two particular problems of this kind are obtained.

Détail

Keywords: Wiener random field of Jech–Chentsov, Wiener cushion, Laplace method in Banach spaces, covariance operator of Gaussian measure, nonlinear Hammerstein equations, high-order hyperbolic boundary-value problems.