局部遍历随机环境下一个重伸缩过程收敛的结果

A Convergence Result of a Rescale Process Within Locally Ergodic Random Environment

考虑一个重伸缩过程(Xη,εt)t≥0,假设{η(x)}x∈Z是由局部遍历性的概率测度分布的,本文研究此过程当ε→0时的极限。证明了在局部遍历性分布条件下,对于R上的二阶连续可微函数f(X)和某个与η独立的齐次扩散函数a(X),这个重伸缩过程依分布με收敛到R上具有无穷小生成元d/dX(a(X)d/dXf(X))的扩散过程。

This paper considers a rescaled process （Xη,εt）t≥0,and it＇s assumed that {η（x）}x∈Z is distributed by a locally ergodic probability measure. The limit of the rescaled process is studied as ε→0. It is proved that under local ergodicity distributions,the rescaled process converges in distribution με to the diffusion process on R with infinitesimal generator d/dX（a（X）d/dXf（X））,for second-order continuous differentiable function f（X） on R and a certain homogenized diffusion function a（X） which is independent of η.