Colloquium @ CEREMADE

Description

Our colloquium takes place on the first Tuesday of each month from 15:30 to 16:30, usually in room A709.


A renowned expert (being an excellent speaker as well) visits us for an afternoon and gives a panorama of one of her research areas. The talk is meant to be accessible to all members of the lab, including PhD students in analysis, game theory, probability and statistics. Ideally, it should start gently with an historical background on the problem and an overview of the main questions and applications, keeping a non technical style during at least the first half of the talk. Of course it is also nice to have a part with more mathematical details: the most appreciated colloquia were those in which the speaker succeeded to develop a nice technical idea or an elegant argument that everyone should know.


Food and drinks are served after the event, usually in Espace 7!


If you know good speakers whom you would love to hear, do not hesitate to suggest their names to the two organizers: Justin Salez and Cristina Toninelli.

Next talk

Date: Tuesday, April 2nd 2024 (15:30-16:30, room A709)


Speaker: Bálint Tóth (University of Bristol and Alfréd Rényi Institute of Mathematics)


Title: The Random Lorentz Gas - Invariance Principle Beyond the Kinetic Time Scales


Abstract: Since the pioneering work of Paul and Tatiana Ehrenfest (1912) the deterministic (Hamiltonian) motion of a point-like particle exposed to the action of a collection of fixed, randomly located short range scatterers has been a much studied model of physical diffusion under fully deterministic (Hamiltonian) dynamics, with random initial conditions. This model of physical diffusion is known under the name of "random Lorentz gas" or "random wind-tree model". Celebrated milestones on the route to better mathematical understanding of this model of true physical diffusion are the Kinetic Limits for the tagged particle trajectory under the so-called Boltzmann-Grad (a.k.a. low density), or weak coupling approximations [Gallavotti (1970), Spohn (1978), Boldrighini-Bunimovich-Sinai (1982), respectively, Kesten-Papanicolaou (1980)]. Once the Kinetic Limits are established, under a second diffusive space-time scaling limit the central limit theorem (CLT) and invariance principle (IP) follows. However, the CLT/IP under bare diffusive space-time scaling (without first applying the kinetic approximations) remains a Holy Grail.


In recent work we have obtained some intermediate results, partially interpolating between the two-steps-limit (first kinetic, then diffusive - as described above) and the bare-diffusive-limit (Holy Grail). We establish the Invariance Principle for the tagged particle trajectories under a joint kinetic+diffusive limiting procedure, performed simultaneously rather than successively, reaching significantly longer time scales than any earlier result. The main ingredient is a coupling of the Hamiltonian trajectory (with random initial conditions) and an approximate Markovized version of the motion, and probabilistic and geometric controls on the efficiency of this coupling. The Holy Grail remains, however, beyond reach.


In the colloquium talk I will present a concise historic survey of the problems and some insight regarding the main ideas of the more recent work.


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Past talks