Brownian snake
= Brownian snake
{c}
{wiki}
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> $(Z_t)_{0\leq t\leq1}$ with <covariance function>
$$
\mathbb E[Z_sZ_t]=m_g(s,t)
=\inf_{r\in[s\wedge t,s\vee t]}g(r).
$$
Consequently $\mathbb E[(Z_t-Z_s)^2]=d_g(s,t)$, the <pseudometric> used by the <real tree encoded by an excursion>.