论文标题
长时间的噪声扩散的大偏差很大
Large Deviations for Small Noise Diffusions Over Long Time
论文作者
论文摘要
我们研究两个问题。首先，我们考虑某些扩散过程的经验度量的较大偏差行为，因为同时，时间范围变得较大，噪声变得很小。在这种渐进状态下，经验度量的大量（LLN）定律由无噪声动力学的独特平衡给出。由于极限噪声的变性，Donsker和Varadhan（1976）的方法不适用，需要新的想法。其次，我们研究了一个缓慢的扩散系统，在该系统中，慢速和快速组件在其自然时间尺度上都具有消失的噪声。这次，LLN受退化平均原理的控制，其中从快速动力学中获得的无噪声系统的局部平衡描述了缓慢分量的渐近演化。我们建立了一个较大的偏差原理，该原则描述了与这种行为的差异概率。一方面，我们的方法比非排定设置需要更强的假设，而另一方面，速率功能采用与非排定同类物具有明显差异的简单和明确的形式。
We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan (1976) are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts.
