Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 220 / 2 / a

For $u\ge 0$, let ${\mathcal{C}}_u$ be the collection of interval components of the superlevel set $\{r:g(r)\ge u\}$. Two times $s,t$ lie in the same member of ${\mathcal{C}}_u$ exactly when $u\le m_g(s,t)$. ThereforeFor arbitrary real ${\lambda }_1,\dots ,{\lambda }_n$, the Tonelli theorem now givesThus $(m_g(s_i,s_j){)}_{i,j}$ is a positive semidefinite matrix.

The head of the Brownian snake driven by $g$ is the centered Gaussian process $(Z_t{)}_{0\le t\le 1}$ with covariance function $\mathbb{E}[Z_s{Z}_t]=m_g(s,t)$. Part i shows that these finite-dimensional distributions exist consistently. Moreover,If $g$ has Hölder constant $L$, then $d_g(s,t)\le 2L|t-s{|}^{\alpha }$. The absolute moment formula for a centered normal distribution consequently gives, for every $p\ge 2$,The Kolmogorov continuity theorem, with $p$ arbitrarily large, produces a modification that is $\gamma$-Hölder continuous for every $\gamma <\alpha /2$. Taking $\gamma =\alpha /2-\epsilon$ proves the claim whenever $0<\epsilon <\alpha /2$; for larger $\epsilon$ the assertion is vacuous.