论文标题
注意ISE的密度和相关扩散
Note on the density of ISE and a related diffusion
论文作者
论文摘要
综合的超青色游览（ISE）是布朗蛇头部空间成分的占用度量，其生命周期过程是归一化的布朗式旅行。这是$ \ mathbb {r} $上的一个随机概率度量，众所周知，它可以描述各种随机离散标记树模型中标签分布的连续限制。 我们表明$ f_ {ise} $，其（随机）密度具有A.S。一个连续且$ \ left（\ frac {1} {2} {2} -a \ right）$ - hölder的衍生$ f'_ {ise} $，用于任何$ a> 0 $，但对于没有$ a <0 $的no $ a <0 $（证明了Bousquet-Mélou和Janson的猜想）。 我们推测$ f_ {ise} $可以表示为表单的二阶扩散 $$ df'_ {ise}（t）= \ sqrt {2f_ {ise}（t）} \，db_t + g \ left（f'_ {ise}（ise}（t），f_ {ise {ise}（ise}（t）（t）（t），\ int _ { - \ int _ { - \ iffty} - \ \ f_ to $ p_ $}连续功能$ g $，以$ t> 0 $，我们在这个方向上提供许多评论和问题。 规律性的证明是基于来自离散树木模型的时刻估计，而扩散的启发式词则来自离散环境中的类似陈述，这是对Bousquet-Mélou的显式产品公式的重新印度和第一作者（2012年）。
The integrated super-Brownian excursion (ISE) is the occupation measure of the spatial component of the head of the Brownian snake with lifetime process the normalized Brownian excursion. It is a random probability measure on $\mathbb{R}$, and it is known to describe the continuum limit of the distribution of labels in various models of random discrete labelled trees. We show that $f_{ISE}$, its (random) density has a.s. a derivative $f'_{ISE}$ which is continuous and $\left(\frac{1}{2}-a\right)$-Hölder for any $a >0$ but for no $a<0$ (proving a conjecture of Bousquet-Mélou and Janson). We conjecture that $f_{ISE}$ can be represented as a second-order diffusion of the form $$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-Mélou and the first author (2012).
