In this thesis, we study large deviations and parameter estimations for small noise diffusion processes. In Chapter 1, we start with the classical limit theorems to intuitively introduce large deviations and parameter estimations, which provide for further developments in the thesis.

The first part, consisting of Chapters 2 - 4, is on large deviations. In Chapter 2, we begin with the simple stochastic differential equation to explain the idea behind the proof of the nonlinear semigroup method, which is used to prove large deviations in Chapters 3 and 4. In the process, viscosity solutions and the Hamilton-Jacobi-Bellman equations are introduced...

We consider a class of slow–fast processes on a connected complete Riemannian manifold M. The limiting dynamics as the scale separation goes to ∞ is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with Hamilton–Jacobi–Bellman (HJB) equation techniques. Our main innovation is solving the comparison principle for viscosity solutions for the HJB equation on M and the construction of a variational viscosity solution for the non-smooth Hamiltonian, which lies at the heart of deriving the action integral representation for the rate function.