报 告 人: Jean-Michel Roquejoffre (Toulouse University)

时间地点: 数学院新楼308，6月11日14:30

题    目: Front propagation in system of mean field game type modelling the diffusion of knowledge

摘   要: The question under study, at large intermediate times, of a system, proposed by the economists Lucas and Moll, aimed at describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy are supposed to share their time between learning and producing. They advance their knowledge by learning from each other and via internal innovation, and their density obeys a forward in time equation of reaction-diffusion type. The learning strategy of the agents is based on the solution to a backward in time nonlocal Hamilton-Jacobi-Bellman equation that is coupled to the equation for the agents density. The result is a system of the mean-field game type.

An important parameter, that measures how successful the learning is, determines different asymptotic regimes. One of them, that does not seem to have been identified in the literature, where most of the agents spend almost all their time to learn, and whose large intermediate time behaviour has a lot to do with Fisher-KPP propgation, will be especially discussed.

个人简介: Jean-Michel Roquejoffre is a Professor at Toulouse University. His research primarily focuses on the large-time behaviour of reaction-diffusion equations in both discrete and continuous media. His work centers on the Fisher-KPP equation, a critical model at the intersection of probability and biological modelling, which is essential for understanding the spatial spread of populations under reproduction and dispersion, as well as the behaviour of the rightmost particle in Branching Brownian Motion.

Through a long-standing collaboration with L. Ryzhik (Stanford University), Professor Roquejoffre has developed new approaches to this model, introducing novel arguments for studying sharp asymptotic behaviour and providing original results on the underlying stochastic processes. He also actively develops mathematical models in ecology and epidemiology.

His investigation into the large-time dynamics of reaction-diffusion fronts has led him to explore several seemingly remote fields. In collaboration with L. Caffarelli, he introduced the notion of nonlocal minimal surfaces, which established significant connections to classical De Giorgi theory and found important applications in image processing. Furthermore, he has made foundational contributions to the analysis of large-time asymptotics in Hamilton-Jacobi equations, where his work includes some of the earliest nontrivial results in the field.