There are two different problems studied in this thesis. The first one is a travelling wave problem. We will improve the result proved in  to derive the ergodic property of the travelling wave behind the wavefront. The second problem is a large deviation problem concerning solutions to certain kind stochastic partial differential equations. We will first briefly introduce some basics about SPDE in chapter 2. In chapter 3, we will prove a large deviation principle for super-Brownian motion when it is considered as a solution to an SPDE, using the LDP for super-Brownian motion when it is considered as a measure-valued branching process as solution to a martingale problem. In chapter 4, we will prove another LDP result for solutions of a stochastic reaction-diffusion equation with degenerate noise term. Finally in chapter 5, we will explore some applications of those LDP results proved previously
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