Editing Brownian tree (section)

=== 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 <math>k</math> leaves numbered from <math>1</math> to <math>k</math>. These trees have <math>2k-1</math> edges with lengths <math>(\ell_1,\dots,\ell_{2k-1})\in \R_+^{2k-1}</math>. A tree is then defined by its shape <math>\tau</math> (which is to say the order of the nodes) and the edge lengths. We define a [[Probability theory|probability law]] <math>\mathbb{P}</math> of a random variable <math>(T,(L_i)_{1\leq i\leq 2k-1})</math> on this space by:{{what|reason = This measure is not normalized|date=July 2023}}

: <math>\mathbb P(T=\tau \,, \, L_i\in [\ell_i, \ell_i + d\ell_i], \forall 1 \leq i \leq 2k-1)= s \exp(-s^2/2)\, d\ell_1 \ldots d\ell_{2k-1}</math>

where <math>\textstyle s = \sum \ell_i</math>.

In other words, <math>\mathbb P</math> depends not on the shape of the tree but rather on the total sum of all the edge lengths.
{{Math theorem
| math_statement = Let <math>X</math> be a random metric space with the tree property, meaning there exists a unique path between two points of <math>X</math>. Equip <math>X</math>  with a probability measure <math>\mu</math>. Suppose the sub-tree of <math>X</math> generated by <math>k</math> points, chosen randomly under <math>\mu</math>, has law <math>\mathbb P</math>. Then <math>X</math> is called a '''Brownian tree'''.
| name = Definition
}}
In other words, the Brownian tree is defined from the laws of all the finite sub-trees one can generate from it.