Brownian tree

 ( Wiener Process ) 

In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a random real tree that can be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three articles published in 1991 and 1993. This tree has since then been generalized.

This random tree has several equivalent definitions and constructions: using sub-trees generated by finitely many leaves, using a Brownian excursion, Poisson separating a straight line or as a limit of Galton-Watson trees.

Intuitively, the Brownian tree is a binary tree whose nodes (or branching points) are Dense set in the tree; which is to say that for any distinct two points of the tree, there will always exist a node between them. It is a fractal object which can be approximated with computers or by physical processes with dendritic structures.

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Definitions The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articles.

David Aldous (1991). 9780521425339 . ISBN 9780521425339

The notions of leaf, node, branch, root are the intuitive notions on a tree (for details, see ).

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Finite-dimensional laws 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:

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

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.

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

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Continuous tree 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 Metric space $d$ on $0,1$ with

$d(x,\; y)\; :=\; e(x)+e(y)-2\; \backslash min\backslash big\backslash \{e(z)\backslash ,\; ;\; z\backslash inx,y\backslash big\backslash \},$ 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 $0,1\backslash ,/\backslash !\backslash sim\_e$.

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

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Poisson line-breaking construction This is also called stick-breaking construction.

Consider a non-homogeneous Poisson point process 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 , the segment $]C\_k,C\_\{k+1\}]$ is similarly glued to the tree $T\_\{k-1\}$, on a uniformly random point of $T\_\{k-1\}$.

This algorithm may be used to simulate numerically Brownian trees.

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Limit of Galton-Watson 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. G_n converges in distribution to a random real tree, which we call a Brownian tree. | name = Definition and Theorem

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).

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