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Pathwise Regularisation of McKean-Vlasov problems

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If you have a question about this talk, please contact Renato Velozo.

McKean—Vlasov equations are a well established modelling tool and area of important mathematical study. A challenging and still unsolved problem is to obtain rigorous results concerning the validity of mean-field approximations to such systems, especially when the interaction is singular.

In this talk I will present a work in preparation by myself and F. Harang (University of Oslo) where we consider a random perturbation of such singular kernels that yields well-posedness of the mean field equation and rigorous mean field limit results for the particle system. The approach is based on the notion of a local-time and encapsulates the physical intuition that at small scales it is extremely unlikely for two particles to be in exactly the same location.

Our approach uses recent ideas from Cattier & Gubinelli ’16 (https://arxiv.org/pdf/1205.1735.pdf), Harang & Perkowski ’20 (https://arxiv.org/pdf/2003.05816.pdf) and Galeati & Gubinelli ’20 (https://arxiv.org/pdf/2003.05816.pdf) to obtain a path-wise regularisation of such equations, combined with the path-wise approach to classical McKean-Vlasov equations presented by Friz et al ’19 (https://arxiv.org/pdf/1812.11773.pdf).

In the talk I will firstly, give a brief introduction to the theory of McKean-Vlasov/mean field problems in general as well as the path-wise regularisation results mentioned above. Then I will explain our new result for regularised particle systems.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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