Long Time Asymptotics of Ornstein-Uhlenbeck Processes in Poisson Random Media

The Models of Random Motions in Random Media (RMRM) have been shown to have fruitful applications in various scientific areas such as polymer physics, statistical mechanics, oceanography, etc. In this dissertation, we consider a special model of RMRM: the Ornstein-Uhlenbeck process in a Poisson random medium and investigate the long time evolution of its random energy. We give complete answers to the long time asymptotics of the exponential moments of the random energy with both positive and negative coefficients, under both quenched and annealed regimes. Through these results, we find out a dramatic difference between the long time behavior of the Brownian motion dynamics and the Ornstein-Uhlenbeck dynamics in the Poisson random medium.