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行为，主要研究了扩散过程在多维随机环境中的递归性。将具有一维稳定过程和形式的环境视为不连续随机环境，将具有一维稳定过程和形式的环境视为连续随机环境，将多维高斯随机环境视为连续随机环境。证明了这些环境下扩散过程的递归性。为了研究扩散在更一般的环境下的长时间渐近行为，用显式速率函数给出了随机积分对应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特征。

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.