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Speaker: Alexandra Holtzinger, University of Oxford
Title: Beyond the classical mean-field scaling: Moderately interacting particles and their fluctuation behaviour
Abstract: In this talk, I will start by giving a short introduction to mean-field limits in general and their limitations when deriving Partial Differential Equations (PDE) by using classical mean-field scalings. In the classical regime, also called weakly interacting particles, the influence of each particle on any other particle scales like 1/N, where N is the number of particles. In the large population limit -- under suitable assumptions -- the particles become independent with a common density function which solves a certain type of PDE, the McKeanVlasov equation. However, this scaling regime does not recover many important partial differential equations from physics and biology, like the well-known porous media equation.
Hence, I will explain how changing the strength of interaction from 1/N to a factor which is still vanishing in the limit for N large, but slightly stronger than 1/N, changes the limiting dynamics. This regime is called moderate interaction since the strength of interaction is stronger than in the weak regime (1/N), but weaker than in the hydrodynamic limit (strongly interacting particles with strength O(1)) and attracted a lot of attention recently.
In addition, I will give insights on recent results on the fluctuations around the mean-field limit for the moderate regime, which can be seen as a Central Limit Theorem for interacting particles. This talk is based on serval articles together with Ansgar Jüngel (TU Wien), Li Chen (University of Mannheim), José Carrillo (University of Oxford) and Shuchen Guo (University of Oxford).