Note on the density of ISE and a related diffusion

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 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 almost surely a derivative f'_ISE which is continuous and (1/2-ε)-Hölder for any ε > 0 but for no ε < 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 (Formula Presented) Mathamatic equation Presented 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).