FKPP equation and fixed points of branching Brownian motion

• Arnab Chowdhury (TIFR CAM, Bengaluru, India)

Room: Amphithéâtre Schwartz We consider the one-dimensional binary branching Brownian motion (BBM) and its fixed point problem. The recent work of Chen, Garban, and Shekhar (PTRF, 2023) classified the fixed points of BBM with the critical drift under the assumption that the fixed points have a top particle (i.e., a finite maximum particle) almost surely. A related result for supercritical drifts was obtained by Kabluchko (J. Appl. Prob., 2012), but under a more restrictive assumption of a locally finite intensity measure. We study the BBM with both critical and supercritical drifts and obtain a complete characterization of the fixed points without any additional assumptions. A key strategy in our analysis is the connection between BBM and the FKPP equation. This talk is based on joint work with Xinxin Chen, Atul Shekhar and Shuo Zhu.

Coexistence for Competing Branching Random Walks with Identical Asymptotic Shape on ℤᵈ

• Partha Pratim Ghosh (Ruhr University Bochum, Germany)

Room: Amphithéâtre Schwartz Abstract: We consider two independent branching random walks that start next to each other on the d-dimensional hypercubic lattice and that carry two different colors. Vertices of the lattice are colored according to the color of the walker cloud that first visits the vertex, leading to the question of possible coexistence in the sense that both colors appear on infinitely many vertices. Under mild conditions, we prove the coexistence for two independently distributed branching random walks obeying the same first- and second-order behavior for their extremal particles. To complement this result, we also exhibit examples for the almost-sure absence of coexistence, for d=1, in cases where the asymptotic shapes of the walker clouds are calibrated to coincide, thereby answering a question by Deijfen and Vilkas (ECP 28(15):1-11, 2023). As a main tool we employ second-order and large-deviation approximations for the position of the extremal particles in one-dimensional branching random walks.

Stochastic Epidemic Model for Malaria: the Law of Large Numbers

• Othmane Baghdadi (Université Mohammed Premier Oujda, Marrocco)

Room: Amphithéâtre Schwartz We study an individual-based stochastic host–vector epidemic model for malaria, in which humans can experience repeated infections over their lifetime. In contrast to classical Ross–Macdonald or compartmental SIR/SEIR models, each infection episode is characterised by a random time-dependent infectivity profile: after infection, a human host transmits parasites to susceptible mosquitoes according to a random infectivity function of the time since infection, while recovered hosts gradually regain susceptibility according to a random susceptibility function. On the vector side, susceptible mosquitoes become infected through contact with infectious humans and then contribute to transmission until death, under a demographic regime that combines birth and mortality processes. We analyse the large-population asymptotic behaviour of this coupled host–vector system and prove a functional law of large numbers (FLLN) by constructing a sequence of i.i.d. auxiliary processes. The limiting dynamics are described by a nonlinear deterministic system of renewal-type integral equations that generalises both the classical Kermack–McKendrick age-of-infection framework and standard malaria models. In this limit, the solution of the limiting deterministic system depends on the expectation of a complicated functional of the random susceptibility functions, but only on the mean infectivity functions of humans and mosquitoes. Othmane Baghdadi¹ and Étienne Pardoux² ¹ Mohammed First University, Oujda, Morocco — othmane.baghdadi.d23@ump.ac.ma ² Aix-Marseille University, CNRS, I2M, 13453 Marseille, France — etienne.pardoux@univ-amu.fr

Finite particle limit in the Ensemble Kalman Sampler

• Louis Carillo (Cermics, Enpc, l'École des Ponts ParisTech)

Room: Amphithéâtre Schwartz Interacting particle systems have attracted increasing interest for sampling and Bayesian inference. They are appealing because they are well suited to parallel implementation and, in several cases, provide derivative-free approximations. In this talk, I focus on the Ensemble Kalman Sampler (EKS), an interacting particle system for sampling that enjoys a property of invariance with respect to affine transformations. EKS evolves an ensemble of particles interacting through the empirical covariance, which acts as an adaptive preconditioner. Although the mean field limit of EKS is relatively well understood, the long-time behavior of the finite particle system remains largely open. We address this gap in the Gaussian setting by first establishing uniform-in-time quantitative bounds on the distance between the finite particle system and its mean field limit. We then derive new estimates on the long-time behavior of the finite particle system, showing that the distance to the target distribution decreases faster with the number of particles than might be expected from the mean field limit alone.

Mean-field limit for the Motach-Tadmor model

• Tamari Kldiashvili (De Vinci Higher Education; University of Graz, Austria)

Room: Amphithéâtre Schwartz In this talk I will focus on the mean-field limit of the Motsch–Tadmor model for flocking birds, which refines the classical Cucker–Smale model through a local normalization. While the mean-field limit is rigorously established for the Cucker–Smale model, the asymmetry of the Motsch–Tadmor interactions breaks the standard tools, leaving the rigorous mean-field limit an open problem. I will present an overview of the model and the problem, together with a propagation of chaos (and hence the mean-field limit) result for the Motsch–Tadmor dynamics, obtained by coupling the N-particle and kinetic flows and controlling their Wasserstein distance via a Gronwall argument.  Joint work with: Stefano Rossi (Sapienza University of Rome) and Lara Trussardi (University of Graz)

Nonlocal Collision Operators: Conservative Forms and Local Entropy Inequalities

• Zhe Chen (MAP5, CNRS, Université Paris Cité)

Room: Amphithéâtre Schwartz The classical Boltzmann equation models dilute gases, where particles are treated as mass points and collisions are local in space. For dense gases and related nonlocal models, such as Enskog, soft-sphere, and Povzner-type equations, collisions couple particles located at different spatial points. This spatial delocalization makes the usual local conservation and entropy structures less direct. In this talk, I will introduce a general framework for delocalized collision operators and explain how their conservation laws can be recovered through suitable spatial and velocity fluxes. I will also present a local entropy inequality and its connection with the H-theorem for dense-gas models.

Finite Element Scheme for Phase Field Model in Two-Phase Flow Computations

• Victoria Iyadunni Ayodele (University of Dundee, Scotland)

Room: Amphithéâtre Schwartz Multiphase flows arise in a wide range of applications, including materials science, petrochemical engineering, and the oil industry. Despite their importance, the accurate numerical simulation of multiphase flows remains challenging due to the complex behaviour of the governing equations and the need for stable long-time computations. The key challenges in simulating multiphase flow include the complexity of the equations, computational resources, accuracy, and stability. To address these challenges, we propose a new Besse- type scheme for the Cahn-Hilliard equation (BSCH) and its coupling with the Navier-Stokes equation (BSCH-NS). The proposed scheme is second order in time and linearly implicit, requiring only the solution of a linear system at each time step. The scheme satisfies the energy dissipation law without imposing any condition on time or mesh size, providing unconditional stability. The key objective of the work is to develop a method that accurately preserves the energy law at the discrete level, and also, a method that shows improved long-time behaviour compared to the existing auxiliary variable method. We study the deformation of an initially rectangular bubble with two immiscible fluids using the proposed scheme, with the interface described by the level curve . The interface evolves under surface tension, driving the phase separation dynamics. The proposed method is compared with the existing auxiliary variable methods to investigate the long-time behaviour of the numerical solution. Numerical experiments are implemented in FreeFEM and MATLAB. The proposed method provides an improved numerical solution that is accurate and stable over a long-time computation, contributing to a better prediction of multiphase flow behaviour in industrial applications. Keywords: Cahn–Hilliard equation, Navier–Stokes equations, Besse-type scheme, two-phase flow, energy dissipation law, unconditional stability, finite element scheme.

Clustering Phenomena in Transformers

• Alexander Kasiman (Technical University Darmstadt)

Room: Amphithéâtre Schwartz

The Zombie Infection Model - A Non-monotone Variant of the SIR Model

• Samuel Modée (University of Bergen, Norway)

Room: Amphithéâtre Schwartz The Zombie Infection Model (ZIM) is a variant of the stochastic SIR model on graphs in which infected nodes are not removed spontaneously, but are instead killed by their susceptible neighbours: a susceptible node becomes infected at rate λ times its number of infected neighbours, while an infected node is removed at rate 1 times its number of susceptible neighbours. This places the ZIM within the class of interacting particle systems, as a hybrid of the SIR and biased-voter (Williams–Bjerknes) models, and produces rich and sometimes counterintuitive behaviour. I will present a first rigorous analysis, focusing on the survival probability — the chance the infection spreads indefinitely — and its monotonicity. On trees this probability is monotone in λ, yet there exist bounded-degree graphs on which it is not. Via couplings with random walks and percolation, we further obtain extinction criteria and survival bounds on complete graphs, regular trees, and Z^d.

Fluctuations of spatial population processes with non-local interactions

• Ruairi Garrett (University of Oxford, UK)

Room: Amphithéâtre Schwartz

Branching-Selection Particle Systems and the F-KPP equation

• Jacob Mercer (University of Oxford, UK)

Room: Amphithéâtre Schwartz We introduce a generalisation of the N-branching Brownian motion process, which is a toy model for the effect of survival of the fittest in a population. We find the hydrodynamic limit of the process and show that it is described by a F-KPP type reaction diffusion equation. We discuss the asymptotic speed of the particle system, the travelling waves of the hydrodynamic limit, and when a so-called `weak selection principle' holds.

From McKean Particle Systems to Branching Feynman-Kac representations: adressing Poisson-Vlasov PDEs

• Daniel Yaacoub (CNRS, LAPLACE (UMR 5213))

Room: Amphithéâtre Schwartz Recent advances have made it possible to develop path-space probabilistic representations of mesoscopic Boltzmann transport nonlinearly coupled to a self-consistent submodel of the force field through forward approaches based on continuous branching stochastic processes. In this work, path-space probabilistic representations of free-space Poisson–Vlasov and Poisson–Boltzmann systems are presented. This yields novel propagator representations and opens new avenues for efficient and benchmark simulations through the use of new Branching Backward Monte Carlo (BBMC) algorithms.

Upper bound on heat kernels of finite particle systems of Keller-Segel type

• Sallah Eddine Boutiah (Université Laval (CA) et L'université de Sétif (Algeria))

Room: Amphithéâtre Schwartz In this talk, I will present upper bounds on the heat kernels of finite-particle systems of Keller–Segel type exhibiting blow-up effects. This is joint work with Damir Kinzebulatov.

Decision-making in heterogeneous self-propelled systems

• Víctor Villegas-Morral (Universitat Politècnica de Catalunya)

Room: Amphithéâtre Schwartz Inspired by empirical observations in animal swarming -- particularly in schooling fish -- we propose an opinion-swarming model for self-propelled particles in order to understand the effect of uninformed individuals in consensus formation. Building on classical bounded-confidence opinion models and self-propelled swarming models, we introduce a three-population framework that distinguishes between leaders, followers, and uninformed individuals. Particles are described by their position, velocity, and a continuous opinion variable; they interact through self-propulsion, alignment, attraction-repulsion forces, and opinion-based mechanisms. First, we derive and analyse an individual-based coupled model that integrates spatial swarming dynamics with the evolution of individual opinions. We perform an extensive numerical study of the relevant parameters and their effect on the dynamics. We derive the mean-field partial differential equations for this coupled individual-based model, which will lay the basis for the study of the long-term behaviour of the system. Our analysis reveals that uninformed individuals, despite lacking any opinion bias, significantly influence group dynamics by diluting the effect of leaders and promoting more democratic decision-making. These findings support the role of uninformed agents in collective decision-making and provide the first analytical insights into leadership and decision-making in heterogeneous crowds. Estrada-Rodriguez, Villegas-Morral, Wolfram (2025) Decision making in heterogeneous self-propelled particle systems https://arxiv.org/abs/2508.06573

A viscosity solutions approach for the study of the mean field limit of the eigenvalues of large random matrices

• Valentin Pesce (Polytechnique)

Room: Amphithéâtre Schwartz In this talk, I shall present how viscosity solutions theory has been recently used to study some mean field limit problems arising in the theory of large random matrices. More precisely, I will present comparison principles both at the level of particles (which represent the eigenvalues of matrices) and at the level of the limit PDE. These principles are robust and are the key points to study the mean field limit of the system of particles. This is a joint work with Charles Bertucci.

Arrhenius-type law and Kramers-type law for a kinetic interacting particles system.

• Hetranso AHNI (Université Jean-Monnet Saint-Étienne)

Room: Amphithéâtre Schwartz In this talk, we discuss the resilience to noise of collective behavior. Our main focus will be on a kinetic particle system developed by Cucker and Smale (2007). We establish key asymptotics for the first exit-time from a safety domain, namely Arrhenius' law and a Kramers-type law, which are well known for non-degenerate diffusions. This is joint work with Jean-François Jabir and Julian Tugaut.

Parametric Inference for particle system driven by fractional noise

• Augustin Puel (Université de Nice)

Room: Amphithéâtre Schwartz