论文标题
在大偏差缩放下随机电流的渐近行为，平均场相互作用和消失的噪声
Asymptotic Behavior of Stochastic Currents under Large Deviation Scaling with Mean Field Interaction and Vanishing Noise
论文作者
论文摘要
我们研究了具有平均场相互作用的扩散颗粒系统的较大偏差行为，该系统通过随机微分方程的集合来描述，其中每个粒子都由消失的独立的布朗尼噪声驱动。描述渐近行为的一个重要对象，因为粒子的数量接近无穷大，而噪声强度接近零，这是在Flandoli等人的意义上与相互作用粒子系统相关的随机电流。 （2005）。我们为粒子系统的路径经验度量以及同时大粒子和小噪声极限的相关随机电流建立了一个关节大偏差原理（LDP）。我们的工作将Orrieri（2018）的最新结果扩展，其中扩散系数被视为身份，到依赖状态的环境，并可能与平均场相互作用，并可能在LDP中允许在ldp的随机频率上具有更强的拓扑拓扑。证明技术与Orrieri（2018）不同，并依赖于随机控制，弱收敛理论和布朗运动拉普拉斯功能的表示公式。
We study the large deviation behavior of a system of diffusing particles with a mean field interaction, described through a collection of stochastic differential equations, in which each particle is driven by a vanishing independent Brownian noise. An important object in the description of the asymptotic behavior, as the number of particles approach infinity and the noise intensity approaches zero, is the stochastic current associated with the interacting particle system in the sense of Flandoli et al. (2005). We establish a joint large deviation principle (LDP) for the path empirical measure for the particle system and the associated stochastic currents in the simultaneous large particle and small noise limit. Our work extends recent results of Orrieri (2018), in which the diffusion coefficient is taken to be identity, to a setting of a state dependent and possibly degenerate noise with the mean field interaction influencing both the drift and diffusion coefficients, and allows for a stronger topology on the space of stochastic currents in the LDP. Proof techniques differ from Orrieri (2018) and rely on methods from stochastic control, theory of weak convergence, and representation formulas for Laplace functionals of Brownian motions.
