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 $f_{ISE}$, its (random) density has a.s. a derivative $f_{ISE}^{\prime }$ which is continuous and $\left(\frac{1}{2}-a\right)$-Hölder for any $a>0$ but for no $a<0$ (proving a conjecture of Bousquet-Mélou and Janson). We conjecture that $f_{ISE}$ can be represented as a second-order diffusion of the form $$d{f}_{ISE}^{\prime }(t)=\sqrt{2f_{ISE}(t)}d{B}_t+g\left(f_{ISE}^{\prime }(t),f_{ISE}(t),{\int }_{-\infty }^t{f}_{ISE}(s)ds\right)dt,$$ 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).