mean-field limit of non-exchangeable interacting diffusions with singular kernels
A talk in the Bielefeld Stochastic Afternoon series by
Xianliang Zhao
| Abstract: | The mean-field limits of interacting diffusions without exchangeability, caused by weighted interactions and non-i.i.d. initial values, are investigated. The weights are mixed, and the interac- tion kernel has a singularity of the type Lq([0,T],Lp(Rd)) for some p,q bigger than one. We demonstrate that the sequence of signed empirical measure processes with arbitary uniform lr-weights, r > 1, weakly converges to a coupled PDE, such as the dynamics describing the passive scalar advected by the 2D Navier-Stokes equation.
Our method is based on a tightness argument and makes use of the systems’ uniform Fisher information. The main difficulty is to determine how to propagate the regularity properties of the limits of empirical measures in the absence of the DeFinetti-Hewitt-Savage theorem for the non-exchangeable case. To this end, a sequence of random measures, which merges weakly with a sequence of weighted empirical measures and has uniform Sobolev regularity, is constructed through the disintegration of the joint laws of particles. This is joint worked with zhenfu wang and rongchan zhu. Within the CRC this talk is associated to the project(s): B1 |