Note on the density of ISE and a related diffusion

Abstract

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 $\mathbb{R}$, 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 $f_{\mathrm{ISE}}$, its (random) density, has almost surely a derivative $f_{\mathrm{ISE}}^{\prime }$ which is continuous and $(\frac{1}{2}-\epsilon )$-Hölder for any $\epsilon >0$ but for no $\epsilon <0$ (proving a conjecture of Bousquet-Mélou and Janson). We conjecture that $f_{\mathrm{ISE}}$ can be represented as a second-order diffusion of the form

for some continuous function $g$, for $t>0$, 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).