Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 201 / 4 / c

The Gaussian process assumption and the covariance formula give a centered Gaussian increment with varianceFor every finite $p>1$, a standard normal variable $Z$ therefore givesFor any $0<\alpha <1/2$, choose $p>\max (2,(1/2-\alpha {)}^{-1})$. Then $\alpha <1/2-1/p$, so parts (a) and (b) give a Hölder continuous function of exponent $\alpha$ as an extension. To obtain one extension with every required exponent, take a countable sequence ${\alpha }_j\uparrow 1/2$ and apply the construction with suitable $p_j$. Intersect the probability-one events. Their continuous extensions agree on the dense set $D$ and hence agree everywhere. For each smaller exponent choose $j$ with $\alpha <{\alpha }_j$; since $|t-s|\le 1$, the bound with ${\alpha }_j$ also implies the bound with $\alpha$. ThusThe exponent constants may depend on $\alpha$ and the sample path. This proves Brownian Hölder regularity for the continuous Gaussian extension without making an uncountable intersection of unrelated full-probability events.