Related papers: The Brownian marble
Arratia, and later T\'oth and Werner, constructed random processes that formally correspond to coalescing one-dimensional Brownian motions starting from every space-time point. We extend their work by constructing and characterizing what we…
The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct…
Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T\'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K.…
The (standard) Brownian web is a collection of coalescing one- dimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is…
Local perturbations of a Brownian motion are considered. As a limit we obtain a non-Markov process that behaves as a reflected Brownian motion on the positive half line until its local time at zero reaches some exponential level, then…
Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations <x(t)x(s)> ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0<H<1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion, while for H…
The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the…
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 construct a family of processes, from a renewal process, that have realizations that converge almost surely to the Brownian motion, uniformly on the unit time interval. Finally we compute the rate of convergence in a particular case.
We consider a branching Brownian motion in $\mathbb{R}^d$. We prove that there exists a random subset $\Theta$ of $\mathbb{S}^{d-1}$ such that the limit of the derivative martingale exists simultaneously for all directions $\theta \in…
We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and…
We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly…
The Brownian web is a random object that occurs as the scaling limit of an infinite system of coalescing random walks. Perturbing this system of random walks by, independently at each point in space-time, resampling the random walk…
In this paper we study the drifted Brownian meander, that is a Brownian motion starting from $ u $ and subject to the condition that $ \min_{ 0\leq z \leq t} B(z)> v $ with $ u > v $. The limiting process for $ u \downarrow v $ is analyzed…
We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the fast…
Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over the whole real line such that B_i(0)=0. We consider the nth iterated Brownian motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of…
We study a system of branching Brownian motions on $\mathbb R$ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. In [7] it has been studied the case of strictly local creations (the new…
Consider $q_n$ a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with $n$ faces. In this paper we show that, when $n$ goes to $+\infty$, $q_n$ suitably normalized converges weakly in a certain sense…
We study the random metric space called the Brownian plane, which is closely related to the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the uniform infinite planar…
Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a…