 Kolmogorov continuity theorem

If a stochastic process indexed by a subset of ${\mathbb{R}}^d$ satisfiesfor some $p,\beta >0$, then it has a Continuous modification. More precisely, that modification is locally $\gamma$-Hölder continuous for every $\gamma <\beta /p$.

The Kolmogorov continuity theorem is a fundamental result in the theory of stochastic processes, particularly in the study of Brownian motion and other continuous-time processes. It provides conditions under which a collection of random variables (typically indexed by time) possesses a continuous version, which means that the sample paths of the process can be modified to be continuous with probability one.