Asymptotics for Brownian motion in Poisson potential with Riesz kernel and in time-independent Gaussian rough noise

In this dissertation, we consider the long time asymptotics for Brownian motion in Poisson random medium or time-independent Gaussian rough medium. We first give answers of the exponential moment asymptotics of Brownian motion in Poisson random medium of a critical case under both quenched and annealed regimes. We then investigate the light tailed case under quenched regime. Finally, we study the quenched large-t asymptotic of Brownian motion in a time-independent Gaussian rough noise.