On the growth of sample paths of Gaussian random fields Consider a centered Gaussian random field on $\mathbb{R}^n$ with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random field?

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*If the random field is assumed in addition to be stationary, it appears, based on the exponential decay of the tail probabilities, that the growth should be logarithmic (as it is in the one-dimensional case) but is that written up somewhere? (The one-dimensional case is treated in Theorem 1.4. in this article: https://www.jstor.org/stable/2239990).


*Without assuming stationarity, I'm interested if the following is true: Given that the covariance has polynomial growth, is it true that sample paths have a.s. polynomial growth?
 A: I will consider stationary
Gaussian processes $X_v$ indexed by $v\in Z^d$, not continuous time (the
argument for continuous time requires a bit of extra work, and some assumptions
on the short-time behavior of the covariance; smoothness of the covariance at $0$ should be enough). In general, good references on such questions are the lecture notes of R. Adler, or his book with Taylor.
By the Royen Gaussian correlation inequality
$$ P(\sup_{v\in Z^d: |v|_\infty<T} |X_v|<R)\geq \prod_{v\in Z^d:|v|_\infty<T} P(|X_v|<R)=  \big(P(|X_0|<R)\big)^{(2T)^d}$$
from which you get a logarithmic behavior (that is, if $R> C_d \sqrt{\log T}$
then the probability above goes to $1$, with explicit $C_d$). The same bound can also be obtained from a union bound. A complimentary lower bound requires more work but with fast enough (any polynomial) decay of correlation, also holds.
A similar argument works for non-stationary fields, but of course the answer depends on the rate of growth of the covariance (and the move from discrete to continuous requires a bit more assumptions). In general, up to extra log factors, the growth of the variance (not covariance, as soon as the latter
decays fast enough, e.g. polynomially) determine the behavior.
