The fluctuation process of the stochastic point vortex model with common noise converges in distribution to the unique strong solution of a linear fluctuation SPDE with multiplicative transport noise and a conditionally Gaussian additive noise.
Mean-field limits of Riesz-type singular flows
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abstract
We provide a proof of mean-field convergence of first-order dissipative or conservative dynamics of particles with Riesz-type singular interaction (the model interaction is an inverse power $s$ of the distance for any $0<s<d$) when assuming a certain regularity of the solutions to the limiting evolution equations. It relies on a modulated-energy approach, as introduced in previous works where it was restricted to the Coulomb and super-Coulombic cases. The method is also capable of incorporating multiplicative noise of transport type into the dynamics. It relies in extending functional inequalities of arXiv:1803.08345, arXiv:2011.12180, arXiv:2003.11704 to more general interactions, via a new, robust proof that exploits a certain commutator structure.
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The fluctuation behaviour of the stochastic point vortex model with common noise
The fluctuation process of the stochastic point vortex model with common noise converges in distribution to the unique strong solution of a linear fluctuation SPDE with multiplicative transport noise and a conditionally Gaussian additive noise.