Mean-field interaction of Brownian occupation measures, I: uniform tube property of the Coulomb functional
Mean-field interaction of Brownian occupation measures, I: uniform tube property of the Coulomb functional
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布朗占位测量的平均场相互作用，I：库仑泛函的均匀管特性
DOI：
10.1214/16-aihp788
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Chiranjib Mukherjee
中科院分区：
文献类型：
--
作者：
W. Koenig;Chiranjib Mukherjee
研究了三维布朗运动经指数倾斜自相互作用与该运动随时间占用量的库仑能的变换路径测度。配分函数的对数渐近性是在20世纪80年代由Donsker和Varadhan [DV83-P]根据变分公式确定的。最近[MV14]引入了一种研究路径测度本身的新技术，它允许证明归一化的职业测度渐近地集中在公式的所有最大值集合周围。在本文中，我们同样证明了占用测度的库仑泛函集中在一致拓扑中最大值的相应库仑泛函集周围。这是在严格证明标准化占领措施收敛到明确的最大化混合的道路上的决定性一步，推导自[BKM15]。我们的方法依赖于导出具有指数小偏差概率的职业测度的库仑泛函的H{o} -连续性，并将[MV14]中发展的大偏差理论应用于职业测度的某个平移不变泛函。
We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, derived in [BKM15]. Our methods rely on deriving H{o}lder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.