完全耦合多尺度弱相互作用粒子系统的中偏差
Moderate deviations for fully coupled multiscale weakly interacting particle systems
AI总结：
本文研究完全耦合双尺度弱相互作用扩散系统在粒子数趋无穷、尺度分离参数趋零联合极限下经验分布的中偏差原理，利用弱收敛方法给出速率函数的有效平均场控制及负Sobolev表示。
摘要：
我们考虑在双尺度环境中运动的一族完全耦合弱相互作用扩散过程。我们研究了当粒子数趋于无穷且时间尺度分离参数同时趋于零的联合极限下，粒子位置经验分布的中偏差原理。我们利用弱收敛方法，该方法以一个有效平均场控制问题的形式，为中偏差速率函数提供了便利表示。我们以适当的“负 Sobolev”形式严格获得了中偏差速率函数的等价表示，这使人联想到 Dawson-Gärtner 于1987年发表的开创性论文中得到的弱相互作用扩散经验测度的大偏差速率函数。在证明过程中，我们获得了相关遍历定理，并严格研究了与 McKean-Vlasov 问题相关的 Poisson 型方程的正则性，这两个主题本身都具有独立研究价值。
英文摘要：
We consider a collection of fully coupled weakly interacting diffusion processes moving in a two-scale environment. We study the moderate deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods, which provide a convenient representation for the moderate deviations rate function in terms of an effective mean field control problem. We rigorously obtain equivalent representations for the moderate deviations rate function in an appropriate "negative Sobolev" form, which is reminiscent of the large deviations rate function for the empirical measure of weakly interacting diffusions obtained in the 1987 seminal paper by Dawson-Gärtner. In the course of the proof we obtain related ergodic theorems and we rigorously study the regularity of Poisson type of equations associated to McKean-Vlasov problems, both of which are topics of independent interest.
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