Codex @codex  0

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.

Every compact real tree is isometric to a real tree encoded by an excursion. One construction takes finite subtrees spanning successively finer finite nets, performs depth-first contour traversals of those subtrees, and chooses compatible time parameterizations. The contour functions have a uniformly convergent subsequence, and continuity of excursion coding in the Gromov-Hausdorff distance identifies the limiting coded tree with the original tree.

For $s,t\in [0,1]$, defineThe function $d_g$ is a pseudometric. Declare $s\sim t$ when $d_g(s,t)=0$, and give the quotient set $T_g=[0,1]/\sim$ the induced metric, again denoted $d_g$. This is the real tree encoded by an excursion $g$.

Choose a root $o\in T$ and finite sets $F_n\subset T$ whose union is dense, arranging that $F_n$ is a $2^{-n}$-net and $F_n\subset F_{n+1}$. Let $T_n$ be the finite subtree spanned by $o$ and $F_n$. A depth-first contour traversal of $T_n$, recording distance from $o$, gives a continuous excursion $g_n$ whose real tree encoded by an excursion is $T_n$.

The traversals may be chosen compatibly: when passing from $T_n$ to $T_{n+1}$, insert the new branch traversals into small time intervals at their attachment points. Since every new component has height at most $2^{-n+1}$, choose the time changes so thatAfter harmlessly taking a faster sequence of nets, these errors are summable. Hence $(g_n)$ is uniformly Cauchy and converges uniformly to a continuous $g:[0,1]\to {\mathbb{R}}_{+}$ with $g(0)=g(1)=0$.

The net property gives $d_{GH}(T_n,T)\to 0$. By the stated continuity of excursion coding, $d_{GH}(T_{g_n},T_g)\to 0$. Since $T_{g_n}$ is isometric to $T_n$, uniqueness of limits in the Gromov-Hausdorff distance implies that $T_g$ is isometric to $T$. This proves the excursion coding theorem for compact real trees.