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)

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

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 Coffe 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.

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 Coffe break

16:00 → 16:15 Student talks

16:15 → 16:30 Student talks

16:30 → 16:45 Student talks

16:45 → 17:00 Student talks

17:00 → 17:15 Student talks

17:15 → 17:30 Student talks

17:30 → 19:30 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 Coffe 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.

12:30 → 14:00 Lunch break

14:00 → 14:15 Student talks

14:15 → 14:30 Student talks

14:30 → 14:45 Student talks

14:45 → 15:00 Student talks

15:00 → 15:15 Student talks

15:15 → 15:30 Student talks

15:30 → 16:00 Coffe 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.

mer. 17 juin

09:00 → 10:30 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.

10:30 → 11:00 Coffe break

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

12:30 → 14:00 Buffet

jeu. 18 juin

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

10:30 → 11:00 Coffe 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 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.

15:30 → 17:00 Course Andreis - Stochastic models for coagulation processes, large scale limits and phase transitions: Exercise session

15:30 → 17:00 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

10:30 → 11:00 Coffe 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