Julien Berestycki
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
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Julien Berestycki has a total of 26 co-authors. Here are the top co-authors by number of shared publications:
- Éric Brunet
LPS-ENS UPMC CNRS 24 rue Lhomond 75231 Cedex 05 Paris France
7 shared publications - Bastien Mallein
Univ Sorbonne Paris Nord, LAGA, UMR 7539, F-93430 Villetaneuse, France
4 shared publications - Cole Graham
University of Wisconsin–Madison, Madison, USA
4 shared publications - Bernard Derrida
École Normale Supérieure
3 shared publications - Yujin H. Kim
Courant Institute, New York University, New York, USA
3 shared publications - Éric Brunet
Laboratoire de Physique de l'École Normale Supérieure ENS Université PSL CNRS Sorbonne Université Université Paris Cité F-75005 Paris France
3 shared publications
Latest Publications
- 3 October 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (11) , 1-53
KPP Traveling Waves in the Half-Space
- 3 October 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (11) , 1-53
KPP Traveling Waves in the Half-Space
- Open Access
The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift
Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift $-ρ$, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten (1978) showed that this process dies out with probability one if and only if $ρ\geq \sqrt{2}$. We show that in the subcritical case when $ρ> \sqrt{2}$, the law of the process conditioned on survival until time $t$ converges as $t \rightarrow \infty$ to a quasi-stationary distribution, which we call the Yaglom limit. We give a construction of this quasi-stationary distribution. We also study the asymptotic behavior as $ρ\downarrow \sqrt{2}$ of this quasi-stationary distribution. We show that the logarithm of the number of particles and the location of the highest particle are of order $ε^{-1/3}$, and we obtain a limit result for the empirical distribution of the particle locations.Open AccessThe Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift
Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift $-ρ$, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten (1978) showed that this process dies out with probability one if and only if $ρ\geq \sqrt{2}$. We show that in the subcritical case when $ρ> \sqrt{2}$, the law of the process conditioned on survival until time $t$ converges as $t \rightarrow \infty$ to a quasi-stationary distribution, which we call the Yaglom limit. We give a construction of this quasi-stationary distribution. We also study the asymptotic behavior as $ρ\downarrow \sqrt{2}$ of this quasi-stationary distribution. We show that the logarithm of the number of particles and the location of the highest particle are of order $ε^{-1/3}$, and we obtain a limit result for the empirical distribution of the particle locations. - Open AccessThe extremal point process of branching Brownian motion in $\mathbb{R}^d$We consider a branching Brownian motion in $\mathbb{R}^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb{R}^d$ of a particle $u$ at time $t$ can be encoded by its direction $\theta^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m^{(d)}_t$ is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on $\mathbb{S}^{d-1} \times \mathbb{R}$. More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to $D_\infty(\theta) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} \theta $, where $D_\infty(\theta)$ is the limit of the derivative martingale in direction $\theta$ and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasi\'nski, Berestycki and Mallein (Ann. Inst. H. Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).Open AccessThe extremal point process of branching Brownian motion in $\mathbb{R}^d$We consider a branching Brownian motion in $\mathbb{R}^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb{R}^d$ of a particle $u$ at time $t$ can be encoded by its direction $\theta^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m^{(d)}_t$ is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on $\mathbb{S}^{d-1} \times \mathbb{R}$. More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to $D_\infty(\theta) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} \theta $, where $D_\infty(\theta)$ is the limit of the derivative martingale in direction $\theta$ and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasi\'nski, Berestycki and Mallein (Ann. Inst. H. Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).
- Open Access
KPP traveling waves in the half-space
We study traveling waves of the KPP equation in the half-space with Dirichlet boundary conditions. We show that minimal-speed waves are unique up to translation and rotation but faster waves are not. We represent our waves as Laplace transforms of martingales associated to branching Brownian motion in the half-plane with killing on the boundary. We thereby identify the waves' asymptotic behavior and uncover a novel feature of the minimal-speed wave $Φ$. Far from the boundary, $Φ$ converges to a logarithmic shift of the 1D wave $w$ of the same speed: $\displaystyle \lim_{y \to \infty} Φ\big(x + \tfrac{1}{\sqrt{2}}\log y, y\big) = w(x)$.Open AccessKPP traveling waves in the half-space
We study traveling waves of the KPP equation in the half-space with Dirichlet boundary conditions. We show that minimal-speed waves are unique up to translation and rotation but faster waves are not. We represent our waves as Laplace transforms of martingales associated to branching Brownian motion in the half-plane with killing on the boundary. We thereby identify the waves' asymptotic behavior and uncover a novel feature of the minimal-speed wave $Φ$. Far from the boundary, $Φ$ converges to a logarithmic shift of the 1D wave $w$ of the same speed: $\displaystyle \lim_{y \to \infty} Φ\big(x + \tfrac{1}{\sqrt{2}}\log y, y\big) = w(x)$. - 16 February 2022
- Published by IOP Publishing in Nonlinearity
- Vol. 35 (4) , 1558-1609
The distance between the two BBM leaders
We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d 12 of this random variable can be expressed in terms of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of P ( d 12 > a ) as a → +∞. These tail asymptotics were previously known to ‘exponential order;’ we discover an algebraic correction to this behavior.
- 16 February 2022
- Published by IOP Publishing in Nonlinearity
- Vol. 35 (4) , 1558-1609
The distance between the two BBM leaders
We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d 12 of this random variable can be expressed in terms of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of P ( d 12 > a ) as a → +∞. These tail asymptotics were previously known to ‘exponential order;’ we discover an algebraic correction to this behavior.
- Open AccessDerivative martingale of the branching Brownian motion in dimension $d \geq 1$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 $\theta \in \Theta$ almost surely. This allows us to define a random measure on $\mathbb{S}^{d-1}$ whose density is given by the derivative martingale. The proof is based on first moment arguments: we approximate the martingale of interest by a series of processes, which do not take into account the particles that travelled too far away. We show that these new processes are uniformly integrable martingales whose limits can be made to converge to the limit of the original martingale.Open AccessDerivative martingale of the branching Brownian motion in dimension $d \geq 1$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 $\theta \in \Theta$ almost surely. This allows us to define a random measure on $\mathbb{S}^{d-1}$ whose density is given by the derivative martingale. The proof is based on first moment arguments: we approximate the martingale of interest by a series of processes, which do not take into account the particles that travelled too far away. We show that these new processes are uniformly integrable martingales whose limits can be made to converge to the limit of the original martingale.
- Open Access
The distance between the two BBM leaders
We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit $d_{12}$ of this random variable can be expressed in terms of PDEs related to the Fisher--KPP equation. We use such a representation to determine the sharp asymptotics of $\mathbb{P}(d_{12} > a)$ as $a\to+\infty$. These tail asymptotics were previously known to "exponential order;" we discover an algebraic correction to this behavior.Open AccessThe distance between the two BBM leaders
We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit $d_{12}$ of this random variable can be expressed in terms of PDEs related to the Fisher--KPP equation. We use such a representation to determine the sharp asymptotics of $\mathbb{P}(d_{12} > a)$ as $a\to+\infty$. These tail asymptotics were previously known to "exponential order;" we discover an algebraic correction to this behavior. - Open Access
A simple backward construction of Branching Brownian motion with large displacement and applications
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation probability of having a large displacement. As an application, we show that Bovier and Hartung (2015)'s results about variable speed branching Brownian motion also describe the extremal point process of branching Ornstein-Uhlenbeck processes.
Open AccessA simple backward construction of Branching Brownian motion with large displacement and applications
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation probability of having a large displacement. As an application, we show that Bovier and Hartung (2015)'s results about variable speed branching Brownian motion also describe the extremal point process of branching Ornstein-Uhlenbeck processes.
- 12 September 2019
- Published by IOP Publishing in Nonlinearity
- Vol. 32 (10) , 3912-3939
Global existence for a free boundary problem of Fisher–KPP type
Motivated by the study of branching particle systems with selection, we establish global existence for the solution of the free boundary problem when the initial condition is non-increasing with as and as . We construct the solution as the limit of a sequence , where each u n is the solution of a Fisher–KPP equation with the same initial condition, but with a different nonlinear term. Recent results of De Masi A et al (2017 (arXiv:1707.00799)) show that this global solution can be identified with the hydrodynamic limit of the so-called N -BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by removing the leftmost particle at each branching event.
- 12 September 2019
- Published by IOP Publishing in Nonlinearity
- Vol. 32 (10) , 3912-3939
Global existence for a free boundary problem of Fisher–KPP type
Motivated by the study of branching particle systems with selection, we establish global existence for the solution of the free boundary problem when the initial condition is non-increasing with as and as . We construct the solution as the limit of a sequence , where each u n is the solution of a Fisher–KPP equation with the same initial condition, but with a different nonlinear term. Recent results of De Masi A et al (2017 (arXiv:1707.00799)) show that this global solution can be identified with the hydrodynamic limit of the so-called N -BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by removing the leftmost particle at each branching event.
- 19 April 2019
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 72 (12) , 2487-2577
Branching Brownian Motion with Decay of Mass and the Nonlocal Fisher‐KPP Equation
In this work we study a nonlocal version of the Fisher‐KPP equation, urn:x-wiley:0010-3640:media:cpa21827:cpa21827-math-0001 and its relation to a branching Brownian motion with decay of mass as introduced in Addario‐Berry and Penington (2015) , i.e., a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in ℝ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighborhood around them (as measured by the function φ).We obtain two types of results. First, we study the behavior of solutions to the partial differential equation above. We show that, under suitable conditions on φ and u0, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher‐KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system.
- 19 April 2019
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 72 (12) , 2487-2577
Branching Brownian Motion with Decay of Mass and the Nonlocal Fisher‐KPP Equation
In this work we study a nonlocal version of the Fisher‐KPP equation, urn:x-wiley:0010-3640:media:cpa21827:cpa21827-math-0001 and its relation to a branching Brownian motion with decay of mass as introduced in Addario‐Berry and Penington (2015) , i.e., a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in ℝ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighborhood around them (as measured by the function φ).We obtain two types of results. First, we study the behavior of solutions to the partial differential equation above. We show that, under suitable conditions on φ and u0, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher‐KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system.