Note on the density of ISE and a related diffusion

TL;DR

This paper investigates the regularity and conjectures a diffusion process for the density of the integrated super-Brownian excursion, linking discrete tree models to continuous stochastic analysis.

Contribution

It proves the Hölder continuity of the density's derivative of ISE and proposes a diffusion representation conjecture based on discrete models.

Findings

The density of ISE has a.s. a continuous derivative with Hölder regularity for any exponent less than 1/2.

The derivative of the density is not Hölder continuous for any exponent greater than 1/2.

A conjectured diffusion process describes the evolution of the density's derivative, connecting discrete and continuous models.

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 $f'_{ISE}$ which is continuous and $\left(\frac{1}{2}-a\right)$-H\"older for any $a >0$ but for no $a<0$ (proving a conjecture of Bousquet-M\'elou and Janson). We conjecture that $f_{ISE}$ can be represented as a second-order diffusion of the form $$df'_{ISE}(t) = \sqrt{2f_{ISE}(t)}\, dB_t + g\left(f'_{ISE}(t), f_{ISE}(t),\int_{-\infty}^t f_{ISE}(s)ds\right)dt,$$ 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-M\'elou and the first author (2012).