2022/10/18 by Guillaume Chapuy, Chapuy, Guillaume, Marckert, Jean-François · 1 citation
Mathematics · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2210.10159
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 fISE, its (random) density has a.s. a derivative f'ISE which is continuous and ((1)/(2)-a)-Hölder for any a >0 but for no a<0 (proving a conjecture of Bousquet-Mélou and Janson). We conjecture that fISE can be represented as a second-order diffusion of the form df'ISE(t) = √2fISE(t) dBt + g(f'ISE(t), fISE(t),∫-∞t fISE(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-Mélou and the first author (2012).