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 $\gamma <\beta /p$.

For a function on the dyadic rationals in $[0,1]$, let $A_n$ be its largest absolute adjacent increment on the level-$n$ grid. If $K_{\alpha }<\infty$ for $\alpha >0$, the function extends uniquely to a Hölder continuous function withChoose $n$ with $2^{-n}\le |t-s|<2^{1-n}$. The level-$n$ left approximations differ by at most two grid steps, and each finer approximation adds at most one level increment. Thus the difference is bounded by $2{\sum }_{j\ge n}{A}_j\le 2^{-n\alpha }{K}_{\alpha }$. For a stochastic process with $\Vert {\xi }_t-{\xi }_s{\Vert }_p\le C|t-s{|}^{\beta }$, the bound $\Vert A_n{\Vert }_p\le C{2}^{-n(\beta -1/p)}$ makes this series converge in the Lp norm for $0<\alpha <\beta -1/p$.