Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions
Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions
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DOI：
10.1007/s10959-020-01071-4
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发表时间：
2021-01
影响因子：
0.8
通讯作者：
A. Budhiraja;Michael Conroy
中科院分区：
文献类型：
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作者：
A. Budhiraja;Michael Conroy
考虑一组粒子，它们的状态演化是通过一个相互作用的扩散系统来描述的，在这个系统中，每个粒子都被一个独立的单独噪声源和所有粒子共有的少量噪声所驱动。粒子之间的相互作用是由于共同的噪声，也通过漂移和扩散系数，取决于国家的经验措施。研究了两种标度下经验测度过程的大偏差行为，一种标度对应于平均场渐近，另一种标度对应于Freidlin-Wentzell小噪声渐近.不同强度的小共同噪声导致不同类型的大偏差行为，我们提供了一个精确的表征的各种制度。速率函数可以被解释为某些随机控制问题的值函数，其中有两种类型的控制;其中一种控制是随机的和非预期的，来自于单个布朗噪声的聚合贡献，而第二种控制是非随机的，对应于影响所有粒子的小的共同布朗噪声。我们还研究了相互作用粒子系统逼近各种类型的Feynman-Kac泛函的大偏差行为。证明是基于布朗运动的指数泛函的随机控制表示和与受控非线性马尔可夫过程相关的随机微分方程弱解的唯一性结果
Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Freidlin–Wentzell small noise asymptotics. Different levels of intensity of the small common noise lead to different types of large deviation behavior, and we provide a precise characterization of the various regimes. The rate functions can be interpreted as the value functions of certain stochastic control problems in which there are two types of controls; one of the controls is random and nonanticipative and arises from the aggregated contributions of the individual Brownian noises, whereas the second control is nonrandom and corresponds to the small common Brownian noise that impacts all particles. We also study large deviation behavior of interacting particle systems approximating various types of Feynman–Kac functionals. Proofs are based on stochastic control representations for exponential functionals of Brownian motions and on uniqueness results for weak solutions of stochastic differential equations associated with controlled nonlinear Markov processes