Note On Gaussian Random Fields \& Underlying Markov Processes Through a Central Limit Theorem

Abstract

Various classes of Gaussian random fields associated with transient Markov processes $Y$ have been introduced in the probability and mathematical physics literature. The present paper is based on a natural class of Gaussian random fields, termed universal Gaussian random fields (UGRF), for an underlying Markov processes $X$, on a state space $(S,\mathcal{S})$ and having an ergodic invariant initial distribution $\ddot{I}€$, via a central limit theorem of Rabi Bhattacharya for appropriately scaled additive integral functionals ${\int }_0^{nt}f(X(s))ds={\sum }_{j=1}^n{\int }_{(j-1)t}^{jt}f(X(s))ds$ for $f\in 1_{\ddot{I}}{€}^{\perp }\equiv \{f\in L^2(S,\ddot{I}€):⟨f,1{⟩}_{\ddot{I}}€=0\}$. A Lamperti-type time change is introduced to obtain an infinite dimensional stationary Ornstein-Uhlenbeck evolution within a framework introduced in a classic paper of K. Itô. In particular it is shown that the Itô's deterministic component vanishes under this time change, and Itô's continuous regularity theory is applied. Connections with GRFs associated with Markov processes $Y$ in a sense of Dynkin, and a sense of Diaconis and Evans, respectively, are established under additional conditions on the infinitesimal generator $(A,\mathcal{D}(A))$ of the underlying Markov process $X$.