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 $π$, 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 $f\in1_π^\perp\equiv \{f\in L^2(S,π):\langle f,1\rangle_π=0\}$. Connections with GRFs associated with Markov processes $Y$ in a sense of Dynkin, and Diaconis and Evans, respectively, are established under additional conditions on the infinitesimal generator $(A,\mathcal{D}(A))$ of the underlying Markov process $X$.