Note on the density of ISE and a related diffusion
Résumé
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 a.s. a derivative $f'_{ISE}$ which is continuous and 1/2-a H\"older 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 $df'_{ISE}(t) = \sqrt{2f_{ISE}(t)}\, dB_t + g (f'_{ISE}(t), f_{ISE}(t),\int_{-\infty}^t f_{ISE}(s)ds )dt,$ 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-M\'elou and the first author (2012).
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