Note on the density of ISE and a related diffusion

  • Guillaume Chapuy

    Université Paris Cité, France
  • Jean-François Marckert

    Université de Bordeaux, Talence, France
Note on the density of ISE and a related diffusion cover
Download PDF

A subscription is required to access this article.


The integrated super-Brownian excursion (ISE) is the occupation measure of the spatial component of the head of the Brownian snake with lifetime process the normalized Brownian excursion. It is a random probability measure on , and it is known to describe the continuum limit of the distribution of labels in various models of random discrete labelled trees. We show that , its (random) density, has almost surely a derivative which is continuous and -Hölder for any but for no (proving a conjecture of Bousquet-Mélou and Janson). We conjecture that can be represented as a second-order diffusion of the form

for some continuous function , for , and we give a number of remarks and questions in that direction. The proof of regularity is based on a moment estimate coming from a discrete model of trees, while the heuristic of the diffusion comes from an analogous statement in the discrete setting, which is a reformulation of explicit product formulas of Bousquet-Mélou and the first author (2012).

Cite this article

Guillaume Chapuy, Jean-François Marckert, Note on the density of ISE and a related diffusion. Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2024), published online first

DOI 10.4171/AIHPD/181