Title: On the exponential ergodicity of the McKean-Vlasov SDE depending on a polynomial interaction Show Chinese title

Title: 关于依赖于多项式相互作用的McKean-Vlasov SDE的指数遍历性

Abstract: In this paper, we study the long time behaviour of the Fokker-Planck and the kinetic Fokker-Planck equations with many body interaction, more precisely with interaction defined by U-statistics, whose macroscopic limits are often called McKean-Vlasov and Vlasov-Fokker-Planck equations respectively. In the continuity of the recent papers [63, [43],[42]] and [44, [74],[75]], we establish nonlinear functional inequalities for the limiting McKean-Vlasov SDEs related to our particle systems. In the first order case, our results rely on large deviations for U-statistics and a uniform logarithmic Sobolev inequality in the number of particles for the invariant measure of the particle system. In the kinetic case, we first prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 ($\mu$) with a rate of convergence which is explicitly computable and independent of the number of particles. In a second time, we quantitatively establish an exponential return to equilibrium in Wasserstein's W 2 --metric for the Vlasov-Fokker-Planck equation. Show Chinese abstract

Abstract: 在本文中，我们研究了具有多体相互作用的Fokker-Planck方程和动力学Fokker-Planck方程的长时间行为，更准确地说，是具有由U统计量定义的相互作用，其宏观极限通常被称为McKean-Vlasov方程和Vlasov-Fokker-Planck方程。 在最近的论文[63, [43],[42]]和[44, [74],[75]]的基础上，我们建立了与我们的粒子系统相关的极限McKean-Vlasov随机微分方程的非线性函数不等式。 在一阶情况下，我们的结果依赖于U统计量的大偏差和粒子系统不变测度在粒子数上的统一对数 Sobolev 不等式。 在动力学情况下，我们首先证明了在加权Sobolev空间H 1 ($\mu$)中，解关于粒子数的均匀指数收敛到平衡状态，收敛速度可以显式计算且与粒子数无关。 其次，我们定量地建立了Vlasov-Fokker-Planck方程在Wasserstein的W 2 -度量下指数返回平衡状态。