The Mean-Field Limit of Stochastic Point Vortex Systems with Multiplicative Noise

Understand

In arXiv:1004.1407, Flandoli, Gubinelli, and Priola proposed a stochastic variant of the classical point vortex system of Helmholtz and Kirchoff in which multiplicative noise of transport-type is added to the dynamics.

• An open problem in the years since is to show that in the mean-field scaling regime, in which the circulations are inversely proportional to the number of vortices, the empirical measure of the system converges to a solution of a two-dimensional Euler vorticity equation with multiplicative noise.
• By developing a stochastic extension of the modulated-energy method of Serfaty and Duerinckx for mean-field limits of deterministic particle systems and by building on ideas introduced by the author for studying such limits at the scaling-critical regularity of the mean-field equation, we solve this problem under minimal assumptions.