= Solution
The head of the <Brownian snake> driven by $g$ is the centered <Gaussian process> $(Z_t)_{0\leq t\leq1}$ with <covariance function> $\mathbb E[Z_sZ_t]=m_g(s,t)$. Part i shows that these <finite-dimensional distribution>[finite-dimensional distributions] exist consistently. Moreover,
$$
\mathbb E[(Z_t-Z_s)^2]
=g(s)+g(t)-2m_g(s,t)=d_g(s,t).
$$
If $g$ has Hölder constant $L$, then $d_g(s,t)\leq2L|t-s|^\alpha$. The <absolute moment> formula for a centered <normal distribution> consequently gives, for every $p\geq2$,
$$
\mathbb E|Z_t-Z_s|^p\leq C_{p,L}|t-s|^{\alpha p/2}.
$$
The <Kolmogorov continuity theorem>, with $p$ arbitrarily large, produces a modification that is $\gamma$-Hölder continuous for every $\gamma<\alpha/2$. Taking $\gamma=\alpha/2-\varepsilon$ proves the claim whenever $0<\varepsilon<\alpha/2$; for larger $\varepsilon$ the assertion is vacuous.
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