Brownian snake

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.