Related papers: Excursion theory for Brownian motion indexed by th…
We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the…
We develop an excursion theory that describes the evolution of a Markov process indexed by a Levy tree away from a regular and instantaneous point $x$ of the state space. The theory builds upon a notion of local time at $x$ that was…
The integrated Brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first…
We consider the model of Brownian motion indexed by the Brownian tree, which has appeared in a variety of different contexts in probability, statistical physics and combinatorics. For this model, the total occupation measure is known to…
We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is…
We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-M\'elou and Janson identifying the distribution of the density at 0 of…
This paper studies Brownian motion subject to the occurrence of a minimal length excursion below a given excursion level. The law of this process is determined. The characterization is explicit and shows by a layer construction how the law…
Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. Only the…
We show that the past and future of half-plane Brownian motion at certain cutpoints are independent of each other after a conformal transformation. Like in Ito's excursion theory, the pieces between cutpoints form a Poisson process with…
In this paper, following earlier results in [2] we derive the asymptotic distribution as $t \to \infty$, of the excursion of Brownian motion straddling $t$, into an interval $(a,b)$, conditional on the event that there is such an excursion.
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 $\mathbb{R}$,…
The Brownian excursion measure is a conformally invariant infinite measure on curves. It figured prominently in one of the first major applications of SLE, namely the explicit calculations of the planar Brownian intersection exponents from…
We derive the distribution of the center of mass $S$ of the integrated superBrownian excursion (ISE) {from} the asymptotic distribution of the Wiener index for simple trees. Equivalently, this is the distribution of the integral of a…
We consider the model of Brownian motion indexed by the Brownian tree. For every $r\geq 0$ and every connected component of the set of points where Brownian motion is greater than $r$, we define the boundary size of this component, and we…
We consider a Brownian tree consisting of a collection of one-dimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the…
By using the law of the excursions of Brownian motion with drift, we find the distribution of the $n-$th passage time of Brownian motion through a straight line $S(t)= a + bt.$ In the special case when $b = 0,$ we extend the result to a…
Excursion reflected Brownian motion (ERBM) is a strong Markov process defined in a finitely connected domain $D \subset \mathbb{C}$ that behaves like a Brownian motion away from the boundary of $D$ and picks a point according to harmonic…
We study the density of the time average of the Brownian meander/excursion over the time interval [0,1]. Moreover we give an expression for the Brownian meander/excursion conditioned to have a fixed time average.
In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian…
We show that local times of super-Brownian motion, or of Brownian motion indexed by the Brownian tree, satisfy an explicit stochastic differential equation. Our proofs rely on both excursion theory for the Brownian snake and tools from the…