On asymptotic behaviors of stochastic processes in random environments and properties of random environments

关于随机环境中随机过程的渐近性态，主要研究了多维随机环境中扩散过程的常返性。作为不连续的随机环境，那些具有1维稳定过程的和的形式被处理和作为连续的随机环境，多维高斯随机环境被考虑。证明了这类环境中扩散过程的常返性。为了研究更一般环境下扩散方程的长时间渐近行为，在显式速率函数下，得到了对应于随机积分的电流长时间行为的大偏差原理.关于随机环境的性质，主要给出了自分解分布的一些刻画。

On asymptotic behaviors of stochastic processes in random environments, mainly the recurrence properties of diffusion processes in multi-dimensional random environments are investigated. As discontinuous random environments, those having the form of sums of 1-dimensional stable processes are treated and as continuous random environments, multi-dimensional Gaussian random environments are considered. The recurrence properties of diffusion processes in those environments are proved. To investigate long time asymptotic behaviors of diffusions in more general environments, the large deviation principle for long time behaviors of current corresponding to stochastic integral is obtained with explicit rate function. On the properties of random environments, mainly some characterizations of self-decomposable distributions are given.