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We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit…
Let $(B(t),\,t\ge0)$ denote the standard, one-dimensional Wiener process and $(\ell(y,t);\, y\in\mathbb{R},\, t\ge0)$ its local time at level $y$ up to time $t$. Then $\big( (B(t),\, \ell(B(t),t)),\; t\ge0 \big)$ is a random path that fills…
We make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a…
We show uniqueness of the spine of a Fleming-Viot particle system under minimal assumptions on the driving process. If the driving process is a continuous time Markov process on a finite space, we show that asymptotically, when the number…
We study the maximal displacement of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered, where the variances of the increments change…
We study the asymptotic behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the…
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…
We study the scaling limit of a branching random walk in static random environment in dimension $d=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension $1$ we characterize the limit as the…
We consider the precise upper large deviations estimates for the maximal displacement of a branching random walk. In addition, we obtain a description of the extremal process of the branching random walk conditioned on this large deviations…
By analyzing matrices involved, we prove that a snapping-out Brownian motion with large permeability coefficients is a good approximation of Walsh's spider process on the star-like graph $K_{1,k}$. Thus, the latter process can be seen as a…
We prove some technical results relating to the Brownian snake on a stable L\'evy tree. This includes some estimates on the range of the snake, estimates on its occupation measure around its minimum and also a proof of the fact that the…
We present new theoretical and empirical results on the probability distributions of species persistence times in natural ecosystems. Persistence times, defined as the timespans occurring between species' colonization and local extinction…
We define a time dependent empirical process based on $n$ i.i.d.~fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for…
We investigate a diffusive motion of a system of interacting Brownian particles in quasi-one-dimensional micropores. In particular, we consider a semi-infinite 1D geometry with a partially absorbing boundary and the hard-core inter-particle…
Consider a branching process $\{Z_n\}$ in a varying environment. Let $\{W_n\}$ be the natural martingale $Z_n/{\bf E}Z_n$. It converges to some random variable $W$ as $n\to\infty$. An important problem is to show that ${\bf P}(W>0)$ equals…
We consider inhomogeneous branching diffusions on an infinite domain of $\mathbb{R}^d$. The first aim of this article is to derive a general criterium under which the size process (number of particles) and the genealogy of the particle…
The random motion of a Brownian particle confined in some finite domain is considered. Quite generally, the relevant statistical properties involve infinite series, whose coefficients are related to the eigenvalues of the diffusion…
We study the large scale behaviour of a population consisting of two types which evolve in dimension d = 1, 2 according to a spatial Lambda- Fleming-Viot process subject to random time-independent selection. If one of the two types is rare…
We consider the diffusion approximation of branching processes in random environment (BPREs). This diffusion approximation is similar to and mathematically more tractable than BPREs. We obtain the exact asymptotic behavior of the survival…
We introduce a general model of trapping for random walks on graphs. We give the possible scaling limits of these Randomly Trapped Random Walks on $\mathbb {Z}$. These scaling limits include the well-known fractional kinetics process, the…