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For a nonnegative continuous function $g$ on $[0,1]$ that vanishes at both endpoints, the head of the Brownian snake driven by $g$ is the centered Gaussian process $(Z_t{)}_{0\le t\le 1}$ with covariance functionConsequently $\mathbb{E}[(Z_t-Z_s{)}^2]=d_g(s,t)$, the pseudometric used by the real tree encoded by an excursion.

If the driving function $g$ is $\alpha$-Hölder continuous, then its Brownian snake has a modification that is $\gamma$-Hölder continuous for every $\gamma <\alpha /2$. Indeed,and the conclusion follows from the Kolmogorov continuity theorem by taking $p$ arbitrarily large.

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.