Brownian tree
In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree 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 are dense 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.
Definitions
The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articles. The notions of leaf, node, branch, root are the intuitive notions on a tree.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 leaves numbered from to. These trees have edges with lengths. A tree is then defined by its shape and the edge lengths. We define a probability law of a random variable on this space by:
where.
In other words, 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.
Continuous tree
The Brownian tree is a real tree defined from a Brownian excursion.Let be a Brownian excursion. Define a pseudometric on with
We then define an equivalence relation, noted on which relates all points such that .
is then a distance on the quotient space.
It is customary to consider the excursion rather than.
Poisson line-breaking construction
This is also called stick-breaking construction.Consider a non-homogeneous Poisson point process with intensity. In other words, for any, is a Poisson variable with parameter. Let be the points of. Then the lengths of the intervals are exponential variables with decreasing means. We then make the following construction:
- The first step is to pick a random point uniformly on the interval. Then we glue the segment to . We obtain a tree with a root, two leaves, as well as one binary branching point.
- At step, the segment is similarly glued to the tree, on a uniformly random point of.
Limit of Galton-Watson trees
Consider a Galton-Watson tree whose reproduction law has finite non-zero variance, conditioned to have nodes. Let be this tree, with the edge lengths divided by. In other words, each edge has length. The construction can be formalized by considering the Galton-Watson tree as a metric space or by using renormalized contour processes.Here, the limit used is the convergence in distribution of stochastic processes in the Skorokhod space or the convergence in distribution defined from the Hausdorff distance.