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