Hölder regularity of the Brownian snake
= Hölder regularity of the Brownian snake
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,
$$
\mathbb E|Z_t-Z_s|^p=C_p d_g(s,t)^{p/2}
\leq C'_p|t-s|^{\alpha p/2},
$$
and the conclusion follows from the <Kolmogorov continuity theorem> by taking $p$ arbitrarily large.