| Abstract: |
| Systems of interacting particles are ubiquitous in physics, biology, chemistry, computer science and the social sciences. In many applications, the desired interaction potential is a highly singular function or distribution, making well-posedness, mean field convergence, and precise fluctuations challenging to obtain. A particularly relevant example are point vortex models of fluids where both singular interactions and non-Gaussian correlations are practically relevant. In this talk I will present recent results, obtained with L. Galeati and K. Le as well as ongoing work with J. Weinberger, on interacting particle systems driven by i.i.d. fractional Brownian motions (fBm), subject to irregular, possibly distributional, pairwise interactions. We show quantitative propagation of chaos and Gaussian fluctuations for these systems, where the singularity of the interaction may be chosen arbitrarily severe provided the noise fluctuates sufficiently fast (in a sense to be made formal). Our proofs rely on a combination of Sznitman`s direct comparison argument with stochastic sewing techniques and U-statistics. Time permitting I will also present complimentary results on multiplicative equations driven by fBm with singular volatitlity and ergodic theory for fBm driven coercive SDEs with singular drift. |
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