Strong Laws of the Processes Generated by Infinite Dimensional Ornstein-Uhlenbeck Processes

Abstract

A real-valued stationary Gaussian process {X(t), − ∞ 0) if EX (t) = 0 and $$\Gamma \left( {s,t} \right) = EX\left( t \right)X\left( s \right) = \left( {\gamma /\lambda } \right)\exp \left( { - \lambda \left| {t - \left. s \right|} \right.} \right).$$ (3.1.1) Let Y(t) = {X 1(t),...X i (t),...}where X i (·)are independent OU processes with coefficients γ i and λ i (i =1,2,...). The infinite-dimensional OU process Y(·)has been extensively studied in the literature of the past twenty or so years with several different applications in mind. For example, it was used to describe physical phenomena subject to random forces in Dawson (1972), and appeared in the infinite-dimensional filtering and quantum string theory in Miyahara (1982) and was also suggested as a model for certain biological systems in Dawson (1972) and Walsh (1981). For a more detailed and accurate discussion along these lines we refer to Antoniadis and Carmona (1987).