Integrated Super-Brownian Excursion (ISE)

• ISE is a canonical random probability measure defined via the occupation measure of a Brownian snake’s head process driven by a normalized excursion.
• Its discrete approximations converge from the vertical profile of random trees to a continuously differentiable density with sharp Hölder regularity properties.
• ISE plays a central role in random geometry and probability, linking discrete combinatorial models with continuous scaling limits and second-order stochastic dynamics.

The integrated super-Brownian excursion (ISE) is a canonical random probability measure on $\mathbb{R}$ that emerges as the scaling limit of various combinatorial and probabilistic models—most notably, the label distributions in large random trees and as the occupation measure of the Brownian snake whose lifetime is a normalized Brownian excursion. The ISE has a density $f_{ISE}$, which is continuous, compactly supported, and almost surely admits a continuous derivative of sharp regularity exponent. Its law plays a central role in the paper of random real trees, continuum random trees, superprocesses, and critical branching structures.

1. Construction of the ISE

ISE is constructed through the occupation measure of the head process of the Brownian snake driven by a standard normalized Brownian excursion $e = (e(s), 0 \leq s \leq 1)$. The height function $c(s) = 2e(s)$ defines the lifetime of the Brownian snake, a process $(W_s, t \in [0, c(s)])$ indexed by $s \in [0,1]$. For each $s$, $W_s$ is a Brownian path of length $c(s)$. The process is such that $W_s$ and $W_{s'}$ coincide up to time $T_{s,s'} = \min\{c(u): u \in [s, s']\}$ and then evolve independently.

The head of the snake is $H_s := W_s(c(s))$, forming a centered Gaussian process over $[0,1]$ with covariance given by $$\operatorname{Cov}(H_s,H_t) = \min\{c(u): u \in [s \wedge t, s \vee t]\}$$. The ISE is the occupation measure of $H$: $$\forall \text{ bounded continuous } g,\quad \int_{\mathbb{R}} g(x) \,\mu_{ISE}(dx) := \int_{0}^{1} g(H_s)\,ds,$$ where $\mu_{ISE}$ is a random probability measure on $\mathbb{R}$ with continuous, compact random support. Absolute continuity with respect to Lebesgue measure holds almost surely, with the Radon-Nikodym density denoted as $f_{ISE} : \mathbb{R} \to \mathbb{R}_+$ such that $\mu_{ISE}(dx) = f_{ISE}(x)dx$ and $\int f_{ISE}(x)\,dx = 1$ (Chapuy et al., 2022).

2. Discrete Approximations and Convergence

ISE arises naturally in the scaling limits of various discrete combinatorial structures. Let $T_n$ denote a uniform random binary (plane) tree with $n$ vertices, where each vertex is assigned an abscissa (number of right edges minus number of left edges from the root). For $k \in \mathbb{Z}$, the random variable $M_k(T_n)$ counts vertices at abscissa $k$. Define the empirical measure,

$$\mu_{T_n} := \frac{1}{n} \sum_{k \in \mathbb{Z}} M_k(T_n)\, \delta_{k/n^{1/4}},$$

which converges in law to $\mu_{ISE}$. More precisely, the "vertical profile" or rescaled label profile process

$$x \mapsto n^{-3/4} M_{\left\lfloor x n^{1/4} \right\rfloor}(T_n)$$

converges in the space $C_0(\mathbb{R})$ to $f_{ISE}$. The tightness of the associated increment process (properly rescaled discrete gradients)

$$t \mapsto n^{-1/2} \Delta_{\lfloor n^{1/4} t \rfloor}(T_n), \quad \Delta_k := M_k - M_{k-1}$$

ensures, via discrete-to-continuum invariance principle, that $f_{ISE}$ is a.s. differentiable, with continuous derivative $f'_{ISE}$ (Chapuy et al., 2022).

3. Regularity, Hölder Exponent, and Non-Smoothness

A key result is the precise regularity of $f_{ISE}$ and its derivative. Using moment-increment bounds for the discrete increments,

$$\mathbb{E}\left[(\Delta_b - \Delta_a)^p\right] \leq C_p n^{p/2} \left(\frac{|b-a|}{n^{1/4}}\right)^{p/2}$$

for even $p \geq 4$, Kolmogorov’s continuity criterion implies that $f'_{ISE}$ is almost surely $(1/2 - \varepsilon)$-Hölder continuous for every $\varepsilon>0$, but for no positive $\varepsilon$ is $f'_{ISE}$ almost surely $(1/2 + \varepsilon)$-Hölder. Thus, $f_{ISE}$ is a.s. not twice differentiable (Chapuy et al., 2022). This sharp regularity exponent fulfills a conjecture of Bousquet-Mélou and Janson.

4. Explicit Distributions and Laplace Functionals

Explicit descriptions are available for several key functionals of ISE. The density at zero, $L_0 = f_{ISE}(0)$, under the excursion measure conditioned on unit total volume, follows a positive $2/3$-stable law,

$L_0 \stackrel{d}{=} 2^{3/4} 3 \, T^{-1/2},$

where $T$ is a positive $2/3$-stable random variable with $\mathbb{E}[e^{-A T}] = \exp(-A^{2/3})$. The explicit density is

$$f_{L_0}(x) = \frac{2^{3/4}\,3}{\Gamma(2/3)}\,x^{-5/3}\exp(-2^{1/2} x^{-2/3}), \quad x>0$$

(Gall et al., 2018). The Laplace functional of $f_{ISE}$ satisfies, for bounded continuous $\varphi$,

$$\mathbb{E}\left[\exp\left(-\int \varphi(x)\, f_{ISE}(x)dx\right)\right] = \mathbb{E}\left[\exp\left(-\int_0^1 \Phi(e(t))dt\right)\right],$$

with $\Phi$ representing the pushforward under local times. The center of mass $M=\int x f_{ISE}(x)dx$ is Gaussian with mean 0, variance $\frac16$; all odd moments vanish and even moments are explicitly computable (Tang et al., 2019).

5. Diffusion Structure and Second-Order SDE

A conjectural stochastic differential equation governs the joint evolution of $(f'_{ISE}(t), f_{ISE}(t),F(t))$ where $F(t):= \int_{-\infty}^t f_{ISE}(s)\,ds$. The proposed form is: $$df'_{ISE}(t) = \sqrt{2 f_{ISE}(t)}\, dB_t + g\left(f'_{ISE}(t), f_{ISE}(t), F(t)\right)\,dt,$$ for $t>0$, where $g$ is a continuous function yet to be made explicit but predicted to manifest as the formal limit of discrete drift terms present in Markov chain approximations of the profile process for large $n$ (Chapuy et al., 2022). This second-order SDE, of "acceleration" type, encapsulates the universal Gaussian fluctuation and algebraic drift structures inherited from underlying tree models.

A related Markov chain structure can be constructed from the discrete vertical profile of plane trees—specifically, a time-homogeneous process $Z_k^* = (\Delta_k^*, M_k^*, S_k^*)$ on $\mathbb{Z}^3$—which, after appropriate scaling, converges to a unique diffusion solution capturing the unconditioned dynamics of $(f'_{ISE}, f_{ISE}, \int f_{ISE})$ (Chapuy et al., 2022). A plausible implication is that the full, bridge-like process under two-sided conditioning would complete the Markovian picture, but rigorous proof remains open.

6. Scaling Limits, Random Geometry, and Applications

ISE emerges as the universal scaling limit for a broad class of critical branching models. In low dimensions ($d \leq 3$), the range and local time of tree-indexed random walk, under proper scaling, converge in distribution to the Lebesgue measure of the support, and the density, of ISE, respectively. The Hausdorff dimension of the support equals 4. Explicit scaling constants relate the discrete occupation-counts and local times to their continuum ISE analogs (Gall et al., 2014).

ISE appears as the law for "jump-shapes" in the limiting occupation density process for ensembles of critical branching random walks (Tang et al., 2019); each jump of the limiting process corresponds, in law, to a scaled copy of the ISE density.

In the context of random planar maps, superprocesses, and the Brownian map, explicit knowledge of ISE functionals (e.g., $f_{ISE}(0)$, joint laws of positive/negative mass) informs the understanding of fractal structures, local time distributions, and geometric level-sets (Gall et al., 2018).

7. Open Problems and Outlook

Central open directions include explicit determination of the function $g$ in the second-order SDE for $f'_{ISE}$, the rigorous construction of conditioned bridge versions of the process, and continuity of conditional laws under "FDBC" (family of distributions given by boundary conditions). Whether the local-limit diffusion approximation holds on bounded intervals conditioned at both endpoints and the extension of these results beyond $d \leq 3$ remain significant research questions (Chapuy et al., 2022). A plausible implication is that further analytic combinatorics, particularly resummation of generating functions, could yield explicit representations for the drift in the ISE diffusion.

ISE continues to serve as a central object in probability, random geometry, and combinatorics, linking continuous and discrete paradigms and providing universal limit laws for a broad array of random structures.