Gaussian processes and Hammerstein integral equations
Martin Pincus · Transactions of the American Mathematical Society · 1968
Introduction. In his thesis M. Schilder [1] has proven an analogue of the classical Laplace asymptotic formula for Weiner integrals.It is the purpose of this paper to generalize this formula to expectations on a large class of Gaussian processes, and to demonstrate a close connection with Hammerstein integral equations.We say that p(a, r), 0^ era r^t, is a covariance function if p(e, r) = p(r, a) and if for any finite set 0 < tx < ■ ■ ■ < rn < t the matrix [p(rt, t,)] is nonnegative definite.A Gaussian process is determined by a covariance function p(a, t), O^crgr^r, and a mean function p(r), Ogr^r.Unless explicitly stated otherwise, we shall assume that the mean function is identically zero.If the covariance function p(a, t) is such that np(o, t)2 dtrdr < oo ) and is positive definite, then it defines a positive definite Hilbert-Schmidt operator A, through the equationwhere L2 is Hubert space of functions x on [0, t] with norm (X,X)X'2= [J^OOrfr]1'2.We shall denote by {kj()} the normalized eigenfunctions and by {pt} the reciprocal eigenvalues, ordered in increasing magnitude, of the operator A.When p(o, t) is continuous and positive definite, and if {<x¡} is a sequence of independent Gaussian random variables with mean 0 and variance 1, then [5, pp.30-34, also §5] there exists a Gaussian process with sample paths x(t), Q-¿T¿t, represented by *»-£&•*» except possibly on a t set of Lebesgue measure 0.