= Solution
Here is the one-parameter <Kolmogorov continuity theorem>. Suppose a real-valued <stochastic process> on $[0,1]$ satisfies, for some $p>0$, $q>0$ and finite $C$,
$$
\mathbb E|X_t-X_s|^p\leq C|t-s|^{1+q},\qquad s,t\in[0,1].
$$
Then it has a <continuous> <modification of a stochastic process> whose paths are <Hölder continuous> of every positive exponent $\gamma<q/p$. A modification <means> equality <almost surely> at each fixed time, not an a priori simultaneous equality at all times. Other compact time intervals follow by rescaling; the vector-valued version uses a <norm> in place of absolute value.
For the proof fix $0<\gamma<q/p$ and use the dyadic grids $D_n=\{j2^{-n}:0\leq j\leq2^n\}$. Let $D_n^*$ be the largest adjacent increment on this grid. The <union bound> and <Markov inequality> give
$$
\mathbb P(D_n^*>2^{-n\gamma})
\leq\sum_{j=0}^{2^n-1}2^{n\gamma p}
\mathbb E|X_{(j+1)2^{-n}}-X_{j2^{-n}}|^p
\leq C2^{-n(q-p\gamma)}.
$$
The right-hand side is summable. The <Borel-Cantelli first lemma> shows that, outside a <null set>, an almost-surely finite random constant $K$ can absorb the finitely many early grids so that
$$
D_n^*\leq K2^{-n\gamma}\quad\text{for every }n.
$$
We now use <dyadic increment chaining>. For two distinct dyadic points $s,t$, choose $n$ with $2^{-(n+1)}<|t-s|\leq2^{-n}$. Approximate them from below by $s_j,t_j\in D_j$, for $j\geq n$. Each refinement moves by zero or one grid step, while $s_n,t_n$ are at most two grid steps apart. The sequences eventually equal the original dyadic points. Telescoping gives
$$
|X_t-X_s|
\leq 2K2^{-n\gamma}+2K\sum_{j=n+1}^{\infty}2^{-j\gamma}
\leq C_\gamma K|t-s|^\gamma,
$$
where a deterministic finite $C_\gamma$ absorbs the <geometric series> and $2^{-n}<2|t-s|$. Thus the sample function on the <dense> union of dyadic grids has a unique Hölder-continuous extension to $[0,1]$. Define this extension to be $\widetilde X$ on the probability-one event, and set it to zero on the exceptional <null set>.
For a fixed $t$, take dyadic $t_j\to t$. The moment condition gives $X_{t_j}\to X_t$ in probability, while the constructed extension gives $X_{t_j}\to\widetilde X_t$ <almost surely>. Uniqueness of a probability limit yields $\widetilde X_t=X_t$ <almost surely>. This proves the required modification property. Performing the argument for a <countable> increasing sequence of exponents tending to $q/p$ gives one version with all smaller positive <Hölder exponents>, since its extensions agree on the dyadic points.
For <Brownian motion>, <Gaussian> increments give, for any integer $r\geq2$,
$$
\mathbb E|B_t-B_s|^{2r}=(2r-1)!!\,|t-s|^r.
$$
Use $p=2r$ and $q=r-1$. The theorem gives every exponent $\alpha<(r-1)/(2r)=1/2-1/(2r)$. For any $0<\alpha<1/2$, choose $r$ large enough. Intersect probability-one events over integer $r$ and compact intervals $[0,N]$ to obtain
$$
\boxed{\text{Brownian paths are locally Hölder continuous of every exponent }0<\alpha<\tfrac12\text{ almost surely}.}
$$
The usual <continuous> Brownian version has the same property: its <continuous> paths agree with the constructed version on a <countable> <dense> set and hence everywhere, outside one <null set>. The Hölder constant depends on the compact interval. This is the precise local meaning of <Brownian Hölder regularity>; it is not a uniform Hölder bound on the entire unbounded time axis.
Back to article page
