Editing Brownian tree (section)

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

Consider a non-homogeneous [[Poisson point process]] {{mvar|N}} with intensity <math>r(t)=t</math>. In other words, for any <math>t>0</math>, <math>N_t</math> is a [[Poisson distribution|Poisson variable]] with parameter <math>t^2</math>. Let <math>C_1, C_2, \ldots</math> be the points of <math>N</math>. Then the lengths of the intervals <math>[C_i,C_{i+1}]</math> are [[Exponential distribution|exponential variables]] with decreasing means. We then make the following construction:

* (''initialisation'') The first step is to pick a random point <math>u</math> [[Continuous uniform distribution|uniformly]] on the interval <math>[0,C_1]</math>. Then we glue the segment <math>]C_1,C_2]</math> to <math>u</math> (mathematically speaking, we define a new distance). We obtain a tree <math>T_1</math> with a root (the point 0), two leaves (<math>C_1</math> and <math>C_2</math>), as well as one binary branching point (the point <math>u</math>).
* (''iteration'') At step {{mvar|k}}, the segment <math>]C_k,C_{k+1}]</math> is similarly glued to the tree <math>T_{k-1}</math>, on a uniformly random point of <math>T_{k-1}</math>.
{{Math theorem
| math_statement = The [[Closure (topology)|closure]] <math>\overline{\bigcup_{k\geq 1}T_k}</math>, equipped with the distance previously built, is called a '''Brownian tree'''.
| name = Definition
}}
This algorithm may be used to simulate numerically Brownian trees.