Abstract: Mean-field limits have become an important tool in order to show convergence of a stochastic particle system to a partial differential equation if the number of particles N becomes large.
From a modeling point of view, it is natural to ask about the error we make if we approximate the particle system by the limiting PDE. Studying the corresponding fluctuations η_N := e_N(μ_N - p) for e_N → ∞ as N → ∞ (where μ_N denotes the random empirical measure process of our particle dynamics and p denotes the solution to the PDE) means studying the 'next order corrections' of our mean-field approximation.
If we can show for a certain scaling of e_N that the fluctuations converge to a Gaussian process this can be seen as analogue to the 'Central Limit Theorem'. In this talk I will give a general introduction to fluctuations in the mean-field setting and explain open problems which we want to solve in an ongoing work.