sample-path continuity criteria for Gaussian processes
Kolmogorov's existence theorem builds a Gaussian process but says nothing about whether its sample paths are continuous, bounded, or hopelessly irregular. The continuity (and boundedness) criteria fill this gap: they give checkable conditions on the covariance, or on the metric entropy, under which a version of the process has continuous paths. For Gaussian processes the answer is unusually clean — continuity is governed entirely by the geometry of the index set in the process's own metric.
There are two standard routes. (1) Kolmogorov's continuity criterion (not specific to Gaussians): if E[ |X_s - X_t|^p ] <= C |s - t|^(1 + beta) for some p, beta > 0, then X has a version whose paths are Holder continuous of any order < beta/p. For a Gaussian process, since E[(X_s - X_t)^2] = d(s,t)^2 and higher moments are powers of the second, this needs only a polynomial bound on the canonical distance d(s, t) = sqrt(E[(X_s - X_t)^2]) and gives Holder paths. (2) The entropy criterion (Dudley, sharp for Gaussians): a centered Gaussian process indexed by a totally bounded (T, d) has a version with bounded, uniformly continuous sample paths whenever Dudley's entropy integral converges, integral_0^infinity sqrt( log N(T, d, epsilon) ) d epsilon < infinity. The same integral that bounds E[sup] also forces continuity, because chaining controls not just the supremum but the modulus of continuity at every scale. For the supremum to even be finite (boundedness), this entropy integral converging is essentially the right condition; Talagrand's majorizing-measure theorem makes the characterization exact.
Why it matters: these criteria are how you justify treating a Gaussian process as a random continuous function (an element of C(T)), which is the prerequisite for Gaussian measures on C(T), for the supremum being measurable, and for Levy's modulus of continuity of Brownian motion. The honest subtleties: continuity is a property of a VERSION (you choose a continuous representative; not every realization in the product-space construction is continuous), and the entropy condition is sufficient but, as a single integral, can fail to be necessary — the exact criterion needs majorizing measures, not just covering numbers.
Brownian motion: E[(B_s - B_t)^2] = |s - t|, so E[|B_s - B_t|^4] = 3|s - t|^2 = 3|s-t|^(1 + 1). Kolmogorov's criterion with p = 4, beta = 1 yields a version with Holder-(< 1/4) paths; pushing the moment exponent up gives Holder of any order < 1/2, the sharp Brownian regularity. Either way, a continuous version exists.
A polynomial moment bound on increments gives Holder-continuous Brownian paths; the entropy integral is the sharper Gaussian criterion.
Continuity is a property of a chosen VERSION, and the entropy integral is sufficient but not necessary; the exact characterization of continuity/boundedness needs majorizing measures.