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

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The paper introduces universal Gaussian random fields (UGRF) for an underlying ergodic Markov process through a central limit theorem, demonstrating their connection to previously studied Gaussian random fields associated with transient Markov processes. A Lamperti-type time change is applied to achieve an infinite-dimensional stationary Ornstein-Uhlenbeck evolution, showing that Itô's deterministic component vanishes under this transformation and establishing connections under specific conditions on the infinitesimal generator of the process.

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arXiv:2606.21759v2 Announce Type: replace 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 $\pi$, 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 $f\in1_\pi^\perp\equiv \{f\in L^2(S,\pi):\langle f,1\rangle_\pi=0\}$. 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. It\^o. In particular it is shown that the It\^o's deterministic component vanishes under this time change, and It\^o'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 $X$.