Summer School EUR MINT 2026 on Particle systems and PDEs

15 juin 2026, 09:00 → 19 juin 2026, 14:00 Europe/Paris

Amphithéâtre Schwartz (Institut de Mathématiques de Toulouse)

Bastien Mallein (Institut de Mathématiques de Toulouse), Elvire Jalran (IMT), Marina Ferreira (CNRS, Université de Toulouse), Pierre Gervais (DMA)

Scope of the school

This summer school introduces fundamental analytical and probabilistic tools used in current research on particle systems. It is primarily aimed at Master's and PhD students, as well as post-doctoral and early career researchers interested in these topics and their connections

The objective of the summer school Particle systems and PDEs is to introduce some of the key analytical and probabilistic tools to study these interacting particles systems; as well as to foster interactions and collaboration between the participants. 

This summer school serves as an introduction to the conference PSPDEs XIV taking place in Toulouse the following week. If you are interested, do not hesitate to sign up as well. 

This event is part of the thematic semester Interacting particles, PDEs and applications, held at Institut de Mathématiques de Toulouse, from April to December 2026.

Content of the school

The school will be centered around four courses, consisting in lectures and exercice sessions, but also contributed talks and a poster session by the participants. Below is the list of courses.

• Stochastic models for coagulation processes, large scale limits and phase transitions
  Luisa Andreis (Università degli Studi di Torino)
• An introduction to mean field kinetic equations
  Francis Filbet ( Université de Toulouse)
• Cumulants, their evolution hierarchies, and how to use them to control chaos for kinetic theory
  Jani Lukkarinen  (University of Helsinki)
• Local and global survival of spatial branching processes and generalized principal eigenvalues
  Pascal Maillard (Université de Toulouse)

Broadcast of the school

Due to technical complications, we have unfortunately decided not to broadcast the school.

Financial support

We offer a limited number of fellowships (housing, conference lunches, and local transport) to support the participation of non-local participants. It is mainly open to Master and PhD students but, exceptionally, recently graduate post-docs can also be considered. If you require such funding, please register before April 1st.

Travelling to Toulouse is not covered by our fellowships. However, support may be considered for plane or train tickets on a case-by-case basis for a few students. 

Acknowledgement

This summer school is organized and funded by the MINT Graduate School.

Poster semester 2026.pdf

Summer School PSPDEs 2026.pdf

Course material

• Francis_notes.pdf
• Francis_talk.pdf
• Francis_talkLight.pdf
• Jani_exercises.pdf
• Jani_Intro_slides.pdf
• Jani_Lect3.pdf
• Jani_Notes.pdf
• Pascal_paper.pdf

lun. 15 juin

09:00 → 10:30 Course Filbet - An Introduction to Mean Field Kinetic Equations: 1/3

This series of lectures explores the fundamental concepts of Mean Field Kinetic Equations, emphasizing both theoretical foundations and practical applications. We begin by examining several classical examples of mean field models, illustrating their relevance across various physical systems. Subsequently, we introduce a comprehensive formalism for the mean field limit within classical mechanics, unifying the diverse models discussed. A key tool employed throughout the analysis is the Monge-Kantorovich distance, which serves as a convenient measure for investigating the stability of the mean field characteristic flow. Central to our approach is Dobrushin's stability theorem, which provides a rigorous framework for establishing the stability of the flow concerning initial conditions in phase space and initial distributions. We demonstrate that the mean field limit of the N-particle system naturally follows from Dobrushin's stability results. Additionally, we utilize this technique to perform numerical analysis of particle methods, offering insights into their stability and convergence properties in the context of mean field dynamics.

10:30 → 11:00 Coffee break

11:00 → 12:30 Course Lukkarinen - Cumulants, their evolution hierarchies, and how to use them to control chaos for kinetic theory: 1/3

Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. In these lectures, we discuss how cumulants can be used to control the magnitude of such independence, using our recent work with Aleksis Vuoksenmaa on the stochastic Kac model [arxiv.org:2407.17068] as a benchmark. We introduce cumulants and Wick polynomials of random variables, and show how they can be used to generate evolution hierarchies for both deterministic and stochastic dynamical systems. Cumulants can measure weak joint dependence accurately, and we explain how this allows controlling the accuracy of kinetic theory in the stochastic Kac model. In that case, almost arbitrary initial data may be studied and we find that the evolution of the system is divided into three regimes: an initial regime of duration of at most O(ln N) for N particles in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime. The techniques involve finding suitable weighted norms and iterative structures for the cumulant hierarchy, such as introducing partition classifiers to index the cumulants.

exercises_Jani.pdf

Toulouse-Lukkarinen-Intro-printout.pdf

Toulouse-Notes-2026_Jani.pdf

12:30 → 14:00 Lunch break

14:00 → 15:30 Course Maillard - Local and global survival of spatial branching processes and generalized principal eigenvalues: 1/3

In this mini-course, I will present an approach to the study of local and global survival of spatial branching processes through generalized principal eigenvalues of linear positive semigroups. The fact that local survival of a branching diffusion process can be characterized via positivity of the generalized principal eigenvalue in the sense of Berestycki-Nirenberg-Varadhan of an associate Schrödinger operator has been established in works by Engländer, Kyprianou and Pinsky in the 2000's. The case of global survival has been studied recently by Oliver Tough and myself. We relate it to several definitions of generalized principle eigenvalues for elliptic operators due to Berestycki and Rossi. In doing so, we extend these notions to fairly general semigroups of linear positive operators. We use this relation to prove new results about the generalized principle eigenvalues, as well as about uniqueness and non-uniqueness of stationary solutions of a generalized FKPP equation. The probabilistic approach through branching processes gives rise to relatively simple and transparent proofs under much more general assumptions, as well as constructions of (counter-)examples to certain conjectures.

P. Maillard, O. Tough (2025) Generalised principal eigenvalues and global survival of branching Markov processes, https://arxiv.org/abs/2505.12127

15:30 → 16:00 Coffee break

16:00 → 16:15 FKPP equation and fixed points of branching Brownian motion

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.

Orateur: Arnab Chowdhury (TIFR CAM, Bengaluru, India)

16:15 → 16:30 Coexistence for Competing Branching Random Walks with Identical Asymptotic Shape on ℤᵈ

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.

Orateur: Partha Pratim Ghosh (Ruhr University Bochum, Germany)

16:30 → 16:45 Stochastic Epidemic Model for Malaria: the Law of Large Numbers

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

Orateur: Othmane Baghdadi (Université Mohammed Premier Oujda, Marrocco)

16:45 → 17:00 Finite particle limit in the Ensemble Kalman Sampler

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.

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

17:00 → 17:15 Mean-field limit for the Motach-Tadmor model

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)

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

17:15 → 17:30 Nonlocal Collision Operators: Conservative Forms and Local Entropy Inequalities

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.

Orateur: Zhe Chen (MAP5, CNRS, Université Paris Cité)

17:30 → 19:30 Welcome reception and Poster session

mar. 16 juin

09:00 → 10:30 Course Filbet - An Introduction to Mean Field Kinetic Equations: 2/3

This series of lectures explores the fundamental concepts of Mean Field Kinetic Equations, emphasizing both theoretical foundations and practical applications. We begin by examining several classical examples of mean field models, illustrating their relevance across various physical systems. Subsequently, we introduce a comprehensive formalism for the mean field limit within classical mechanics, unifying the diverse models discussed. A key tool employed throughout the analysis is the Monge-Kantorovich distance, which serves as a convenient measure for investigating the stability of the mean field characteristic flow. Central to our approach is Dobrushin's stability theorem, which provides a rigorous framework for establishing the stability of the flow concerning initial conditions in phase space and initial distributions. We demonstrate that the mean field limit of the N-particle system naturally follows from Dobrushin's stability results. Additionally, we utilize this technique to perform numerical analysis of particle methods, offering insights into their stability and convergence properties in the context of mean field dynamics.

10:30 → 11:00 Coffee break

11:00 → 12:30 Course Lukkarinen - Cumulants, their evolution hierarchies, and how to use them to control chaos for kinetic theory: 2/3

Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. In these lectures, we discuss how cumulants can be used to control the magnitude of such independence, using our recent work with Aleksis Vuoksenmaa on the stochastic Kac model [arxiv.org:2407.17068] as a benchmark. We introduce cumulants and Wick polynomials of random variables, and show how they can be used to generate evolution hierarchies for both deterministic and stochastic dynamical systems. Cumulants can measure weak joint dependence accurately, and we explain how this allows controlling the accuracy of kinetic theory in the stochastic Kac model. In that case, almost arbitrary initial data may be studied and we find that the evolution of the system is divided into three regimes: an initial regime of duration of at most O(ln N) for N particles in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime. The techniques involve finding suitable weighted norms and iterative structures for the cumulant hierarchy, such as introducing partition classifiers to index the cumulants.

exercises_Jani.pdf

Toulouse-Lukkarinen-Intro-printout.pdf

Toulouse-Notes-2026_Jani.pdf

12:30 → 14:00 Lunch break

14:00 → 14:15 Finite Element Scheme for Phase Field Model in Two-Phase Flow Computations

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.

Orateur: Victoria Iyadunni Ayodele (University of Dundee, Scotland)

14:15 → 14:30 Clustering Phenomena in Transformers

Orateur: Alexander Kasiman (Technical University Darmstadt)

14:30 → 14:45 The Zombie Infection Model - A Non-monotone Variant of the SIR Model

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.

Orateur: Samuel Modée (University of Bergen, Norway)

14:45 → 15:00 Fluctuations of spatial population processes with non-local interactions

Orateur: Ruairi Garrett (University of Oxford, UK)

15:00 → 15:15 Branching-Selection Particle Systems and the F-KPP equation

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.

Orateur: Jacob Mercer (University of Oxford, UK)

15:15 → 15:30 From McKean Particle Systems to Branching Feynman-Kac representations: adressing Poisson-Vlasov PDEs

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.

Orateur: Daniel Yaacoub (CNRS, LAPLACE (UMR 5213))

15:30 → 16:00 Coffee break

16:00 → 17:30 Course Filbet - An Introduction to Mean Field Kinetic Equations: Exercise session

This series of lectures explores the fundamental concepts of Mean Field Kinetic Equations, emphasizing both theoretical foundations and practical applications. We begin by examining several classical examples of mean field models, illustrating their relevance across various physical systems. Subsequently, we introduce a comprehensive formalism for the mean field limit within classical mechanics, unifying the diverse models discussed. A key tool employed throughout the analysis is the Monge-Kantorovich distance, which serves as a convenient measure for investigating the stability of the mean field characteristic flow. Central to our approach is Dobrushin's stability theorem, which provides a rigorous framework for establishing the stability of the flow concerning initial conditions in phase space and initial distributions. We demonstrate that the mean field limit of the N-particle system naturally follows from Dobrushin's stability results. Additionally, we utilize this technique to perform numerical analysis of particle methods, offering insights into their stability and convergence properties in the context of mean field dynamics.

16:00 → 17:30 Course Lukkarinen - Cumulants, their evolution hierarchies, and how to use them to control chaos for kinetic theory: Exercise session

Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. In these lectures, we discuss how cumulants can be used to control the magnitude of such independence, using our recent work with Aleksis Vuoksenmaa on the stochastic Kac model [arxiv.org:2407.17068] as a benchmark. We introduce cumulants and Wick polynomials of random variables, and show how they can be used to generate evolution hierarchies for both deterministic and stochastic dynamical systems. Cumulants can measure weak joint dependence accurately, and we explain how this allows controlling the accuracy of kinetic theory in the stochastic Kac model. In that case, almost arbitrary initial data may be studied and we find that the evolution of the system is divided into three regimes: an initial regime of duration of at most O(ln N) for N particles in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime. The techniques involve finding suitable weighted norms and iterative structures for the cumulant hierarchy, such as introducing partition classifiers to index the cumulants.

exercises_Jani.pdf

Toulouse-Lukkarinen-Intro-printout.pdf

Toulouse-Notes-2026_Jani.pdf

mer. 17 juin

09:00 → 10:30 Course Lukkarinen - Cumulants, their evolution hierarchies, and how to use them to control chaos for kinetic theory: 3/3

Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. In these lectures, we discuss how cumulants can be used to control the magnitude of such independence, using our recent work with Aleksis Vuoksenmaa on the stochastic Kac model [arxiv.org:2407.17068] as a benchmark. We introduce cumulants and Wick polynomials of random variables, and show how they can be used to generate evolution hierarchies for both deterministic and stochastic dynamical systems. Cumulants can measure weak joint dependence accurately, and we explain how this allows controlling the accuracy of kinetic theory in the stochastic Kac model. In that case, almost arbitrary initial data may be studied and we find that the evolution of the system is divided into three regimes: an initial regime of duration of at most O(ln N) for N particles in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime. The techniques involve finding suitable weighted norms and iterative structures for the cumulant hierarchy, such as introducing partition classifiers to index the cumulants.

exercises_Jani.pdf

Toulouse-Lukkarinen-Intro-printout.pdf

Toulouse-Notes-2026_Jani.pdf

10:30 → 11:00 Coffee break

11:00 → 12:30 Course Andreis - Stochastic models for coagulation processes, large scale limits and phase transitions: 1/3

This series of lectures introduces stochastic models for coagulation and their connections with kinetic equations, random graphs, and phase transitions. We begin with the classical Marcus–Lushnikov process, a finite-particle Markov model in which clusters merge at rates prescribed by a coagulation kernel, and show how the Smoluchowski coagulation equation emerges as its law-of-large-numbers limit. Special attention will be given to some kernels which are exactly solvable, and will serve as guiding examples throughout the course. In particular, the multiplicative kernel which provides a fundamental bridge between coagulation theory and random graph processes.

A central theme will be the phenomenon of gelation, where mass is lost from the limiting kinetic equation and, at the particle level, macroscopic clusters appear in finite time. We discuss how this phase transition can be understood both analytically, through moment estimates and mass conservation, and probabilistically.

The lectures will also explore extensions beyond the classical space homogeneous setting, including cluster-valued, inhomogeneous, and spatial coagulation models. Along the way, we emphasize the probabilistic tools used to study these systems: martingale formulations, hydrodynamic limits, random graphs, with a special focus on large-deviation estimates. The goal is to provide a unified introduction to stochastic coagulation models, from foundational mean-field limits to current research questions on gelation, criticality, and spatial inhomogeneity.

12:30 → 14:00 Buffet

14:00 → 14:15 Steady States and Hypocoercivity for a Nonlinear Perturbation of an Inhomogeneous Nonlinear Boltzmann Equation

Building on the BGK fixed-point approach of Evans and Menegaki, we study kinetic models for gases driven by a spatially varying heat bath. We extend this viewpoint using infinite-time maps and linear bath estimates to obtain existence and stability results for BGK and perturbative Boltzmann models. Computed stationary profiles illustrate how the resulting density and temperature fields respond to the imposed thermostat.

Orateur: Alexander Pilakoutas (Warwick University)

14:15 → 14:30 Upper bound on heat kernels of finite particle systems of Keller-Segel type

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.

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

14:30 → 14:45 Decision-making in heterogeneous self-propelled systems

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

Orateur: Víctor Villegas-Morral (Universitat Politècnica de Catalunya)

14:45 → 15:00 A viscosity solutions approach for the study of the mean field limit of the eigenvalues of large random matrices

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.

Orateur: Valentin Pesce (Polytechnique)

15:00 → 15:15 Arrhenius-type law and Kramers-type law for a kinetic interacting particles system.

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.

Orateur: Hetranso AHNI (Université Jean-Monnet Saint-Étienne)

15:15 → 15:30 Parametric Inference for particle system driven by fractional noise

Orateur: Augustin Puel (Université de Nice)

jeu. 18 juin

09:00 → 10:30 Course Andreis - Stochastic models for coagulation processes, large scale limits and phase transitions: 2/3

This series of lectures introduces stochastic models for coagulation and their connections with kinetic equations, random graphs, and phase transitions. We begin with the classical Marcus–Lushnikov process, a finite-particle Markov model in which clusters merge at rates prescribed by a coagulation kernel, and show how the Smoluchowski coagulation equation emerges as its law-of-large-numbers limit. Special attention will be given to some kernels which are exactly solvable, and will serve as guiding examples throughout the course. In particular, the multiplicative kernel which provides a fundamental bridge between coagulation theory and random graph processes.

A central theme will be the phenomenon of gelation, where mass is lost from the limiting kinetic equation and, at the particle level, macroscopic clusters appear in finite time. We discuss how this phase transition can be understood both analytically, through moment estimates and mass conservation, and probabilistically.

The lectures will also explore extensions beyond the classical space homogeneous setting, including cluster-valued, inhomogeneous, and spatial coagulation models. Along the way, we emphasize the probabilistic tools used to study these systems: martingale formulations, hydrodynamic limits, random graphs, with a special focus on large-deviation estimates. The goal is to provide a unified introduction to stochastic coagulation models, from foundational mean-field limits to current research questions on gelation, criticality, and spatial inhomogeneity.

10:30 → 11:00 Coffee break

11:00 → 12:30 Course Maillard - Local and global survival of spatial branching processes and generalized principal eigenvalues: 2/3

In this mini-course, I will present an approach to the study of local and global survival of spatial branching processes through generalized principal eigenvalues of linear positive semigroups. The fact that local survival of a branching diffusion process can be characterized via positivity of the generalized principal eigenvalue in the sense of Berestycki-Nirenberg-Varadhan of an associate Schrödinger operator has been established in works by Engländer, Kyprianou and Pinsky in the 2000's. The case of global survival has been studied recently by Oliver Tough and myself. We relate it to several definitions of generalized principle eigenvalues for elliptic operators due to Berestycki and Rossi. In doing so, we extend these notions to fairly general semigroups of linear positive operators. We use this relation to prove new results about the generalized principle eigenvalues, as well as about uniqueness and non-uniqueness of stationary solutions of a generalized FKPP equation. The probabilistic approach through branching processes gives rise to relatively simple and transparent proofs under much more general assumptions, as well as constructions of (counter-)examples to certain conjectures.

P. Maillard, O. Tough (2025) Generalised principal eigenvalues and global survival of branching Markov processes, https://arxiv.org/abs/2505.12127

12:30 → 14:00 Lunch break

14:00 → 15:30 Course Filbet - An Introduction to Mean Field Kinetic Equations: 3/3

This series of lectures explores the fundamental concepts of Mean Field Kinetic Equations, emphasizing both theoretical foundations and practical applications. We begin by examining several classical examples of mean field models, illustrating their relevance across various physical systems. Subsequently, we introduce a comprehensive formalism for the mean field limit within classical mechanics, unifying the diverse models discussed. A key tool employed throughout the analysis is the Monge-Kantorovich distance, which serves as a convenient measure for investigating the stability of the mean field characteristic flow. Central to our approach is Dobrushin's stability theorem, which provides a rigorous framework for establishing the stability of the flow concerning initial conditions in phase space and initial distributions. We demonstrate that the mean field limit of the N-particle system naturally follows from Dobrushin's stability results. Additionally, we utilize this technique to perform numerical analysis of particle methods, offering insights into their stability and convergence properties in the context of mean field dynamics.

15:30 → 16:00 Coffee break

16:00 → 17:30 Course Andreis - Stochastic models for coagulation processes, large scale limits and phase transitions: Exercise session

This series of lectures introduces stochastic models for coagulation and their connections with kinetic equations, random graphs, and phase transitions. We begin with the classical Marcus–Lushnikov process, a finite-particle Markov model in which clusters merge at rates prescribed by a coagulation kernel, and show how the Smoluchowski coagulation equation emerges as its law-of-large-numbers limit. Special attention will be given to some kernels which are exactly solvable, and will serve as guiding examples throughout the course. In particular, the multiplicative kernel which provides a fundamental bridge between coagulation theory and random graph processes.

A central theme will be the phenomenon of gelation, where mass is lost from the limiting kinetic equation and, at the particle level, macroscopic clusters appear in finite time. We discuss how this phase transition can be understood both analytically, through moment estimates and mass conservation, and probabilistically.

The lectures will also explore extensions beyond the classical space homogeneous setting, including cluster-valued, inhomogeneous, and spatial coagulation models. Along the way, we emphasize the probabilistic tools used to study these systems: martingale formulations, hydrodynamic limits, random graphs, with a special focus on large-deviation estimates. The goal is to provide a unified introduction to stochastic coagulation models, from foundational mean-field limits to current research questions on gelation, criticality, and spatial inhomogeneity.

16:00 → 17:30 Course Maillard - Local and global survival of spatial branching processes and generalized principal eigenvalues: Exercise session

In this mini-course, I will present an approach to the study of local and global survival of spatial branching processes through generalized principal eigenvalues of linear positive semigroups. The fact that local survival of a branching diffusion process can be characterized via positivity of the generalized principal eigenvalue in the sense of Berestycki-Nirenberg-Varadhan of an associate Schrödinger operator has been established in works by Engländer, Kyprianou and Pinsky in the 2000's. The case of global survival has been studied recently by Oliver Tough and myself. We relate it to several definitions of generalized principle eigenvalues for elliptic operators due to Berestycki and Rossi. In doing so, we extend these notions to fairly general semigroups of linear positive operators. We use this relation to prove new results about the generalized principle eigenvalues, as well as about uniqueness and non-uniqueness of stationary solutions of a generalized FKPP equation. The probabilistic approach through branching processes gives rise to relatively simple and transparent proofs under much more general assumptions, as well as constructions of (counter-)examples to certain conjectures.

P. Maillard, O. Tough (2025) Generalised principal eigenvalues and global survival of branching Markov processes, https://arxiv.org/abs/2505.12127

ven. 19 juin

09:00 → 10:30 Course Andreis - Stochastic models for coagulation processes, large scale limits and phase transitions: 3/3

This series of lectures introduces stochastic models for coagulation and their connections with kinetic equations, random graphs, and phase transitions. We begin with the classical Marcus–Lushnikov process, a finite-particle Markov model in which clusters merge at rates prescribed by a coagulation kernel, and show how the Smoluchowski coagulation equation emerges as its law-of-large-numbers limit. Special attention will be given to some kernels which are exactly solvable, and will serve as guiding examples throughout the course. In particular, the multiplicative kernel which provides a fundamental bridge between coagulation theory and random graph processes.

A central theme will be the phenomenon of gelation, where mass is lost from the limiting kinetic equation and, at the particle level, macroscopic clusters appear in finite time. We discuss how this phase transition can be understood both analytically, through moment estimates and mass conservation, and probabilistically.

The lectures will also explore extensions beyond the classical space homogeneous setting, including cluster-valued, inhomogeneous, and spatial coagulation models. Along the way, we emphasize the probabilistic tools used to study these systems: martingale formulations, hydrodynamic limits, random graphs, with a special focus on large-deviation estimates. The goal is to provide a unified introduction to stochastic coagulation models, from foundational mean-field limits to current research questions on gelation, criticality, and spatial inhomogeneity.

10:30 → 11:00 Coffee break

11:00 → 12:30 Course Maillard - Local and global survival of spatial branching processes and generalized principal eigenvalues: 3/3

In this mini-course, I will present an approach to the study of local and global survival of spatial branching processes through generalized principal eigenvalues of linear positive semigroups. The fact that local survival of a branching diffusion process can be characterized via positivity of the generalized principal eigenvalue in the sense of Berestycki-Nirenberg-Varadhan of an associate Schrödinger operator has been established in works by Engländer, Kyprianou and Pinsky in the 2000's. The case of global survival has been studied recently by Oliver Tough and myself. We relate it to several definitions of generalized principle eigenvalues for elliptic operators due to Berestycki and Rossi. In doing so, we extend these notions to fairly general semigroups of linear positive operators. We use this relation to prove new results about the generalized principle eigenvalues, as well as about uniqueness and non-uniqueness of stationary solutions of a generalized FKPP equation. The probabilistic approach through branching processes gives rise to relatively simple and transparent proofs under much more general assumptions, as well as constructions of (counter-)examples to certain conjectures.

P. Maillard, O. Tough (2025) Generalised principal eigenvalues and global survival of branching Markov processes, https://arxiv.org/abs/2505.12127