Brownian tree

The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articles.{{Cite journal |last=Aldous |first=David |date=1991 |title=The Continuum Random Tree I |journal=The Annals of Probability |volume=19 |issue=1 |pages=1–28|doi=10.1214/aop/1176990534 |doi-access=free }}{{Cite journal |last=Aldous |first=David |date=1991-10-25 |title=The continuum random tree. II. An overview |url=https://books.google.com/books?id=FerdFlyRS8oC&dq=info:arqXCCYZRZAJ:scholar.google.com&pg=PA23 |journal=Stochastic Analysis |volume=167 |pages=23–70|doi=10.1017/CBO9780511662980.003 |isbn=9780521425339 }}{{Cite journal |last=Aldous |first=David |date=1993 |title=The Continuum Random Tree III |journal=The Annals of Probability |volume=21 |issue=1 |pages=248–289 |doi=10.1214/aop/1176989404 |jstor=2244761 |s2cid=122616896 |issn=0091-1798|doi-access=free }} The notions of leaf, node, branch, root are the intuitive notions on a tree (for details, see real trees).

This definition gives the finite-dimensional laws of the subtrees generated by finitely many leaves.

Let us consider the space of all binary trees with $k$ leaves numbered from $1$ to $k$. These trees have $2k-1$ edges with lengths $(\backslash ell\_1,\backslash dots,\backslash ell\_\{2k-1\})\backslash in\; \backslash R\_+^\{2k-1\}$. A tree is then defined by its shape $\backslash tau$ (which is to say the order of the nodes) and the edge lengths. We define a probability law $\backslash mathbb\{P\}$ of a random variable $(T,(L\_i)\_\{1\backslash leq\; i\backslash leq\; 2k-1\})$ on this space by:{{what|reason = This measure is not normalized|date=July 2023}}

: $\backslash mathbb\; P(T=\backslash tau\; \backslash ,,\; \backslash ,\; L\_i\backslash in\; [\backslash ell\_i,\; \backslash ell\_i\; +\; d\backslash ell\_i],\; \backslash forall\; 1\; \backslash leq\; i\; \backslash leq\; 2k-1)=\; s\; \backslash exp(-s^2/2)\backslash ,\; d\backslash ell\_1\; \backslash ldots\; d\backslash ell\_\{2k-1\}$

where $\backslash textstyle\; s\; =\; \backslash sum\; \backslash ell\_i$.

In other words, $\backslash mathbb\; P$ depends not on the shape of the tree but rather on the total sum of all the edge lengths.

{{Math theorem

| math_statement = Let $X$ be a random metric space with the tree property, meaning there exists a unique path between two points of $X$. Equip $X$ with a probability measure $\backslash mu$. Suppose the sub-tree of $X$ generated by $k$ points, chosen randomly under $\backslash mu$, has law $\backslash mathbb\; P$. Then $X$ is called a Brownian tree.

| name = Definition

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In other words, the Brownian tree is defined from the laws of all the finite sub-trees one can generate from it.

The Brownian tree is a real tree defined from a Brownian excursion (see characterisation 4 in Real tree).

Let $e=(e(x),0\backslash leq\; x\backslash leq\; 1)$be a Brownian excursion. Define a pseudometric $d$ on $[0,1]$ with

: $d(x,\; y)\; :=\; e(x)+e(y)-2\; \backslash min\backslash big\backslash \{e(z)\backslash ,\; ;\; z\backslash in[x,y]\backslash big\backslash \},$ for any $x,y\backslash in\; [0,1]$

We then define an equivalence relation, noted $\backslash sim\_e$ on $[0,1]$ which relates all points $x,y$ such that $d(x,y)=0$ .

: $x\backslash sim\_e\; y\; \backslash Leftrightarrow\; d(x,y)=0.$

$d$ is then a distance on the quotient space $[0,1]\backslash ,/\backslash !\backslash sim\_e$.

{{Math theorem

| math_statement = The random metric space $\backslash big([0,1]\backslash ,/\backslash !\backslash sim\_e,\backslash ,\; d\backslash big)$ is called a Brownian tree.

| name = Definition

}}

It is customary to consider the excursion $e/2$ rather than $e$.

This is also called stick-breaking construction.

Consider a non-homogeneous Poisson point process {{mvar|N}} with intensity $r(t)=t$. In other words, for any $t>0$, $N\_t$ is a Poisson variable with parameter $t^2$. Let $C\_1,\; C\_2,\; \backslash ldots$ be the points of $N$. Then the lengths of the intervals $[C\_i,C\_\{i+1\}]$ are exponential variables with decreasing means. We then make the following construction:

• (initialisation) The first step is to pick a random point $u$ uniformly on the interval $[0,C\_1]$. Then we glue the segment $]C\_1,C\_2]$ to $u$ (mathematically speaking, we define a new distance). We obtain a tree $T\_1$ with a root (the point 0), two leaves ($C\_1$ and $C\_2$), as well as one binary branching point (the point $u$).
• (iteration) At step {{mvar|k}}, the segment $]C\_k,C\_\{k+1\}]$ is similarly glued to the tree $T\_\{k-1\}$, on a uniformly random point of $T\_\{k-1\}$.

{{Math theorem

| math_statement = The closure $\backslash overline\{\backslash bigcup\_\{k\backslash geq\; 1\}T\_k\}$, equipped with the distance previously built, is called a Brownian tree.

| name = Definition

}}

This algorithm may be used to simulate numerically Brownian trees.

Consider a Galton-Watson tree whose reproduction law has finite non-zero variance, conditioned to have $n$ nodes. Let $\backslash tfrac\{1\}\{\backslash sqrt\{n\}\}G\_n$ be this tree, with the edge lengths divided by $\backslash sqrt\{n\}$. In other words, each edge has length $\backslash tfrac\{1\}\{\backslash sqrt\{n\}\}$. The construction can be formalized by considering the Galton-Watson tree as a metric space or by using renormalized contour processes.

{{Math theorem

| math_statement = $\backslash frac\{1\}\{\backslash sqrt\{n\}\}G\_n$ converges in distribution to a random real tree, which we call a Brownian tree.

| name = Definition and Theorem

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Here, the limit used is the convergence in distribution of stochastic processes in the Skorokhod space (if we consider the contour processes) or the convergence in distribution defined from the Hausdorff distance (if we consider the metric spaces).

{{Reflist}}

Category:Wiener process

Category:Fractals