The Brownian snake and random trees

3 May 2004
Svante Janson
The Brownian snake (with lifetime given by a normalized Brownian excursion) arises as a natural limit when studying random trees. This may be used in both directions, i.e. to obtain asymptotic results for random trees in terms of the Brownian snake, or, conversely, to deduce properties of the Brownian snake from asymptotic properties of random trees. The arguments are based on Aldous' theory of the continuum random tree. I will discuss two such situations: 1. The Wiener index of random trees converges, after suitable scaling, to the integral (=mean position) of the head of the Brownian snake. This enables us to calculate the moments of this integral. 2. A branching random walk on a random tree converges, after suitable scaling, to the Brownian snake, provided the distribution of the increments does not have too large tails. For i.i.d increments Y with mean 0, a necessary and sufficient condition is that the tails are o(y^{-4}); in particular, a finite fourth moment is enough, but weaker moment conditions are not.
  • Stochastic Analysis Seminar