On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions

Understand

The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential.

• We focus our attention on the combined mean field and diffusive(homogenisation)limits.
• In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say if it admits more than one steady state.
• A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators.