奇異擾動擴散的漸進常態

Abstract

給定一個隨機擴散過程。這個擴散過程具有兩個時間尺度。一個是改變極快速的尺度，而另一個是比較慢的尺度。在本論文裡，我們有興趣的是擴散過程的停留時間之函數當ε →0。在我們的直覺中，當ε →0我們認為這個擴散過程會被比較快的部份所控制。為了使我們的直覺更加明確，我們使用這個擴散過程的機率密度函數的逼近式去估計當ε →0時的行為。用以這個擴散過程的機率密度函數的逼近式，我們將證明這個擴散過程的停留時間之函數的大數法則以及漸進常 態。

Let Xε (·) be a diffusion process satisfying. This diffusion process has two time scales. One is a rapidly changing scale, and the other is a slowly varying scale. In this paper, we are interested in a function of the occupation time of when ε → 0. In our intuition, we think this diffusion will be driven by its fast part when ε → 0. To make our intuition more precisely, we use the asymptoticity for the density of this diffusion to estimate its behavior when ε →0. By virtue of asymptoticity for the density of this diffusion, we will show the law of large numbers and the asymptotic normality of a function of the occupation time of this process.