论文标题
完全耦合的多尺度弱相互作用粒子系统的中等偏差
Moderate deviations for fully coupled multiscale weakly interacting particle systems
论文作者
论文摘要
我们考虑在两个规模环境中移动的完全耦合弱相互作用的扩散过程的集合。我们研究了粒子位置在粒子数量生长到无穷大的情况下，粒子位置的经验分布的中等偏差原理，并且时间尺度的分离参数同时为零。我们利用弱收敛方法，该方法在有效的平均场控制问题方面为中度偏差率函数提供了方便的表示。我们严格地以适当的“负Sobolev”形式获得了中等偏差速率函数的等效表示，这让人联想到Dawson-Gärtner在1987年在1987年开创性论文中获得的弱相互作用扩散的经验度量的较大偏差率函数。在证据的过程中，我们获得了相关的千古定理，并严格研究了与麦基恩·维拉索夫（McKean-Vlasov）问题相关的泊松类型方程的规律性，这两者都是独立关注的主题。
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.
