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Bastien Mallein

Publications and source records attributed to Bastien Mallein.

At least 55 records · Page 3Linked to original sources
arXiv2018

On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

Consider a branching random walk on the real line. Madaule showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen proved that the renormalized trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probability for the trajectories of two individuals selected according to the Gibbs measure to have split before time $t$, partially answering a question of Derrida and Spohn.

math.PR↗
arXiv2018

Capturability-based Pattern Generation for Walking with Variable Height

Capturability analysis of the linear inverted pendulum (LIP) model enabled walking with constrained height based on the capture point. We generalize this analysis to the variable-height inverted pendulum (VHIP) and show how it enables 3D walking over uneven terrains based on capture inputs. Thanks to a tailored optimization scheme, we can compute these inputs fast enough for real-time model predictive control. We implement this approach as open-source software and demonstrate it in dynamic simulations.

cs.RO↗
arXiv2018

Infinitely ramified point measures and branching Lévy processes

We call a random point measure infinitely ramified if for every $n\in \mathbb N$, it has the same distribution as the $n$-th generation of some branching random walk. On the other hand, branching Lévy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some Lévy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and Lévy processes: the value at time $1$ of a branching Lévy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time $1$ of some branching Lévy process.

math.PR↗
arXiv2018

Biggins' Martingale Convergence for Branching Lévy Processes

A branching Lévy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet $(σ^2,a,Λ)$, where the branching Lévy measure $Λ$ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $(σ^2,a,Λ)$ for additive martingales to have a non-degenerate limit.

math.PR↗
arXiv2018

Brownian motion and Random Walk above Quenched Random Wall

We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_n\}$ and $\{W_n\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N} B_n \geq W_n|W) = N^{-γ+ o(1)}$ for a non-random $γ\geq 1/2$. In the classical setting, $W_n \equiv 0$, it is well-known that $γ= 1/2$. We prove that for any non-trivial $W$ one has $γ>1/2$ and the exponent $γ$ depends only on $\text{Var}(B_1)/\text{Var}(W_1)$. Our result holds also in the continuous setting, when $B$ and $W$ are independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck processes. In the latter case the probability decays at exponential rate.

math.PR↗
arXiv2018

Genealogy of the extremal process of the branching random walk

The extremal process of a branching random walk is the point measure recording the position of particles alive at time $n$, shifted around the expected position of the minimal position. Madaule proved that this point measure converges, as $n \to \infty$, toward a randomly shifted, decorated Poisson point process. In this article, we study the joint convergence of the extremal process together with its genealogical informations. This result is then used to describe the law of the decoration in the limiting process, as well as to study the supercritical Gibbs measures of the branching random walk.

math.PR↗
arXiv2018

$N$-Branching random walk with $α$-stable spine

We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida. In this model the size of the population is fixed to a constant $N$. At each step individuals in the population reproduce independently, making children around their current position. Only the $N$ rightmost children survive to reproduce at the next step. Bérard and Gouéré studied the speed at which the cloud of individuals drifts, assuming the tails of the displacement decays at exponential rate; Bérard and Maillard took interest in the case of heavy tail displacements. We take interest in an intermediate model, considering branching random walks in which the critical spine behaves as an $α$-stable random walk.

math.PR↗
arXiv2018

Branching random walk with selection at critical rate

We consider a branching-selection particle system on the real line. In this model the total size of the population at time $n$ is limited by $\exp\left(a n^{1/3}\right)$. At each step $n$, every individual dies while reproducing independently, making children around their current position according to i.i.d. point processes. Only the $\exp\left(a(n+1)^{1/3}\right)$ rightmost children survive to form the $(n+1)^\mathrm{th}$ generation. This process can be seen as a generalisation of the branching random walk with selection of the $N$ rightmost individuals, introduced by Brunet and Derrida. We obtain the asymptotic behaviour of position of the extremal particles alive at time $n$ by coupling this process with a branching random walk with a killing boundary.

math.PR↗
arXiv2018

Branching-stable point measures and processes

We introduce and study the class of branching-stable point measures, which can be seen as an analog of stable random variables when the branching mechanism for point measures replaces the usual addition. In contrast with the classical theory of stable (Lévy) processes, there exists a rich family of branching-stable point measures with \emph{negative} scaling exponent, which can be described as certain Crump-Mode-Jagers branching processes. We investigate the asymptotic behavior of their cumulative distribution functions, that is, the number of atoms in $(-\infty, x]$ as $x\to \infty$, and further depict the genealogical lineage of typical atoms. For both results, we rely crucially on the work of Biggins.

math.PR↗
arXiv2018

Balance control using both ZMP and COM height variations: A convex boundedness approach

Developments for 3D control of the center of mass (CoM) of biped robots are currently located in two local minima: on the one hand, methods that allow CoM height variations but only work in the 2D sagittal plane; on the other hand, nonconvex direct transcriptions of centroidal dynamics that are delicate to handle. This paper presents an alternative that controls the CoM in 3D via an indirect transcription that is both low-dimensional and solvable fast enough for real-time control. The key to this development is the notion of boundedness condition, which quantifies the capturability of 3D CoM trajectories.

cs.RO↗
arXiv2018

The genealogy of an exactly solvable Ornstein-Uhlenbeck type branching process with selection

We study the genealogy of a solvable population model with $N$ particles on the real line which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around $a$ times its current position, where $a>0$ is a parameter of the model. Then, the $N$ rightmost new-born children are selected to form the next generation. We show that the genealogical trees of the process converge to those of a Beta coalescent as $N \to \infty$. The process we consider can be seen as a toy-model version of a continuous-time branching process with selection, in which particles move according to independent Ornstein-Uhlenbeck processes. The parameter $a$ is akin to the pulling strength of the Ornstein-Uhlenbeck motion.

math.PR↗
arXiv2018

Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk

Consider a supercritical branching random walk on the real line. The consistent maximal displacement is the smallest of the distances between the trajectories followed by individuals at the $n$th generation and the boundary of the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that under some integrability conditions, the consistent maximal displacement grows almost surely at rate $λ^* n^{1/3}$ for some explicit constant $λ^*$. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.

math.PR↗
arXiv2017

Maximal displacement in a branching random walk through interfaces

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.

math.PR↗
arXiv2017

Second order behavior of the block counting process of beta coalescents

The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg proved a law of large numbers for this quantity. Recently, Limic and Talarczyk proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-time branching processes.

math.PR↗
arXiv2016

A $N$-branching random walk with random selection

We consider an exactly solvable model of branching random walk with random selection, which describes the evolution of a population with $N$ individuals on the real line. At each time step, every individual reproduces independently, and its offspring are positioned around its current locations. Among all children, $N$ individuals are sampled at random without replacement to form the next generation, such that an individual at position $x$ is chosen with probability proportional to $\mathrm{e}^{βx}$. We compute the asymptotic speed and the genealogical behavior of the system.

math.PR↗
arXiv2016

Exercices sur les temps locaux de semi-martingales continues et les excursions browniennes

Depuis le tout début du XX${}^\text{e}$ siècle, l'étude des processus stochastiques est un domaine très actif de la recherche en mathématiques. Parmi ces processus, le mouvement brownien --- dont l'étude mathématique a été initiée dès 1900, avec la thèse de Bachelier, entre autres travaux --- a joué, et joue encore, un rôle primordial. Ceci peut s'expliquer par le fait que le mouvement brownien est l'objet limite quasi-universel qui apparaît dans le théorème central limite, lorsqu'on fait agir le temps. Depuis la fin de la seconde guerre mondiale et les travaux d'Itô, Meyer, Tanaka et bien d'autres, les temps locaux et les excursions sont devenus des outils essentiels pour étudier ce processus. Les exercices de ce volume ont été élaborés, année après année, par le second auteur, soit à la suite de lectures d'articles présentant, parfois avec des méthodes très différentes, telle ou telle propriété brownienne, soit simplement pour illustrer le contenu de son cours de DEA (anciennement), de M2 aujourd'hui. Le premier auteur en a organisé la synthèse, de façon économique et néanmoins --- espérons-le --- très lisible. Les chapitres ont été conçus pour créer un aller-retour permanent entre les principaux résultats du cours et les exercices corrigés, afin que la compréhension des uns renforce celle des autres. C'est ainsi que de nombreuses solutions d'exercices données ici offrent un aperçu de la façon de prouver certains des théorèmes rappelés plus haut.

math.PR↗
arXiv2016

Asymptotic of the maximal displacement in a branching random walk

In this article, we study the maximal displacement in a branching random walk. We prove that its asymptotic behaviour consists in a first almost sure ballistic term, a negative logarithmic correction in probability and stochastically bounded fluctuations. This result, proved by Addario-Berry and Reed, and Hu and Shi is given here under close-to-optimal integrability conditions. We provide simple proofs for this result, also deducing the genealogical structure of the individuals that are close to the maximal displacement.

math.PR↗