Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 34 / 3 / b

First construct a centred Gaussian process on the countable dyadic set with covariance function $\mathbb{E}(B_s{B}_t)=\min (s,t)$. These consistent finite-dimensional normal distributions exist because, for coefficients $a_j$,the Kolmogorov extension theorem realizes them on one probability space. Every increment has the normal distribution $N(0,|t-s|)$, so for every finite $p\ge 1$,Use the preceding estimate with $\beta =1/2$, $p>2$ and $0<\alpha <1/2-1/p$. It yields an almost surely Hölder continuous extension from the dense dyadic set to $[0,1]$. Approximate arbitrary times by dyadic times; continuity of the paths and of the Gaussian covariance function preserves all finite-dimensional laws. Disjoint increments are jointly Gaussian with zero cross-covariances, hence independent, with the required variances. Together with $B_0=0$ and continuity, this constructs Brownian motion.

For any $\alpha <1/2$, choose $p$ large enough. Taking a countable increasing sequence of exponents tending to $1/2$ gives simultaneous Hölder regularity of every order $\alpha <1/2$ on a common probability-one set; the same argument on each compact time interval gives local Brownian Hölder regularity. This method does not give the endpoint exponent. In fact that endpoint fails: if a path had a finite $1/2$-Hölder seminorm $K$, all its $2^n$ adjacent normalized increments at level $n$ would have magnitude at most $K$. These are independent standard normal random variables, so for a fixed integer $m$ the probability that their maximum is at most $m$ is $\mathbb{P}(|Z|\le m{)}^{2^n}\to 0$. The event of a finite $K$ is contained in the union over $m$ of events with this bound at every level, each of probability zero. Thus Brownian paths are not $1/2$-Hölder continuous on $[0,1]$ almost surely.