Mean‐Field Interaction of Brownian Occupation Measures II: A Rigorous Construction of the Pekar Process
Mean‐Field Interaction of Brownian Occupation Measures II: A Rigorous Construction of the Pekar Process
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布朗占据测度的平均场相互作用 II：Pekar 过程的严格构造
DOI：
10.1002/cpa.21682
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发表时间：
2015
影响因子：
3
通讯作者：
Chiranjib Mukherjee
中科院分区：
文献类型：
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作者：
E. Bolthausen;W. Koenig;Chiranjib Mukherjee
我们考虑了三维相互作用布朗路径上对应于吉布斯测度的平均场相互作用。相互作用是自吸引的，由奇异库仑势给出。该模型的配分函数的对数渐近性在20世纪80年代由Donsker和Varadhan根据Pekar变分公式确定，这与强耦合下极化子问题的配分函数的行为一致。基于此，Spohn在1986年提出了一个启发性的观察，即极化子路径测度的强耦合行为，在某些时间尺度上，应该类似于一个称为Pekar过程的过程，其分布可以从感兴趣的平均场测度的极限渐近行为中猜测出来，其严格的分析仍然开放。本文致力于精确分析这些平均场路径测度和归一化占领测度向Pekar变分问题的最大化者的显式混合的收敛性。这导致了上述Pekar过程的严格构造，因此有助于在路径测度水平上理解极化子问题的“平均场近似”。我们的证明方法是基于Mukherjee和Varadhan在2016年开发的紧致大偏差理论; König和Mukherjee在2015年将其扩展到奇异库仑相互作用的一致强度量，以及Bolthausen和Schmock在1997年的工作中出现的部分路径交换论点的启发。
We consider mean‐field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self‐attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan in terms of the Pekar variational formula, which coincides with the behavior of the partition function of the polaron problem under strong coupling. Based on this, in 1986 Spohn made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the Pekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean‐field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean‐field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence is a contribution to the understanding of the “mean‐field approximation” of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed by Mukherjee and Varadhan in 2016; its extension to the uniform strong metric for the singular Coulomb interaction was carried out by König and Mukherjee in 2015, as well as an idea inspired by a partial path exchange argument appearing in 1997 in work by Bolthausen and Schmock.© 2017 Wiley Periodicals, Inc.
布朗运动与二维库仑势的相互作用
DOI：
--
发表时间：
2009
期刊：
影响因子：
--
作者：
長田博文
通讯作者：
長田博文