Related papers: The distance between the two BBM leaders
We study the dynamics of the outliers for a large number of independent Brownian particles in one dimension. We derive the multi-time joint distribution of the position of the rightmost particle, by two different methods. We obtain the two…
In this paper, we consider a Fisher-KPP equation with an advection term and two free boundaries, which models the behavior of an invasive species in one dimension space. When spreading happens (that is, the solution converges to a positive…
We first study a $d$-dimensional branching Brownian motion (BBM) among mild Poissonian obstacles, where a random trap field in $\mathbb{R}^d$ is created via a Poisson point process. The trap field consists of balls of fixed radius centered…
In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two…
We study the height of the maximal particle at time $t$ of a one dimensional branching Brownian motion with a space-dependent branching rate. The branching rate is set to zero in finitely many intervals (obstacles) of order $t$. We obtain…
We consider branching Brownian motion in which initially there is one particle at $x$, particles produce a random number of offspring with mean $m+1$ at the time of branching events, and each particle branches at rate $\beta = 1/2m$.…
We take the point of view of a particle performing random walk with bounded jumps on $\mathbb{Z}^d$ in a stationary and ergodic random environment. We prove the quenched large deviation principle (LDP) for the pair empirical measure of the…
The question how the extremal values of a stochastic process achieved on different time intervals are correlated to each other has been discussed within the last few years on examples of the running maximum of a Brownian motion, of a…
We investigate the transport of Brownian particles in a two-dimensional potential under the action of a uniform external force. The potential is periodic in one direction and confines the particle to a narrow channel of varying…
We present an alternative to the well-known Anderson's formula for the probability that a first exit time from the planar region between two slopping lines -a_1 t -b_1 and a_2 t + b_2 by a standard Brownian motion is greater than T. As the…
The logarithmic correction for the order of the maximum of a two-type reducible branching Brownian motion on the real line exhibits a double jump when the parameters (the ratio of the diffusion coefficients of the two types of particles,…
In this work we study a non-local version of the Fisher-KPP equation, \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\tfrac{1}{2}\Delta u +u (1- \phi \ast u), \quad t>0, \quad x\in \mathbb{R}, u(0,x)=u_0(x), \quad x\in…
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 investigate the transport of inertial particles by cellular flows when advection dominates over inertia and diffusion, that is, for Stokes and P\'eclet numbers satisfying $\mathrm{St} \ll 1$ and $\mathrm{Pe} \gg 1$. Starting from the…
We study the large-time behaviour of Brownian particles moving through a viscous medium in a confined potential, and which are further subjected to position-dependent driving forces that are periodic in time. We focus on the case where…
The first of $N$ identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target, sets the time scale of activation, which in general is much faster than the arrival to the target of only a single…
The paper studies asymptotic properties of estimators of multidimensional stochastic differential equations driven by Brownian motions from high-frequency discrete data. Consistency and central limit properties of a class of estimators of…
Consider n non-intersecting Brownian motions on $\mathbb{R}$, depending on time $t \in [0,1]$, with $m_i$ particles forced to leave from $a_i$ at time $t=0$, $1\leq i\leq q$, and $n_j$ particles forced to end up at $b_j$ at time $t=1$,…
We consider random temporal graphs, a version of the classical Erd\H{o}s--R\'enyi random graph G(n,p) where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time…
Consider the continuous greedy paths model: given a $d$-dimensional Poisson point process with positive marks interpreted as masses, let $\mathrm P(\ell)$ denote the maximum mass gathered by a path of length $\ell$ starting from the origin.…