Asymptotics of Large Deviations for Wiener Random Fields in Lp-Norm, Nonlinear Hammerstein Equations, and High-Order Hyperbolic Boundary-Value Problems

Abstract

This paper provides derivation, for $1 u \Biggr\} $$ for two Gaussin fields, namely, the Wiener field of Jech--Chentsov and the so-called cushion, being, respectively, the multiparameter analogues of the Wiener process and the Brownian bridge. These Gaussian fields have zero means, and their respective covariance functions are $$ \prod_{i=1}^n \min (t_i, s_i) \quad\mbox{and}\quad \prod_{i=1}^n [\min (t_i, s_i) - t_i s_i] , \qquad t = (t_1, \dots, t_n),\quad 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~${\bf R}^n $ and the hyperbolic boundary-value problems of high order. Solutions of two particular problems of this kind are obtained.