The Moderate Deviations Of SPDEs And Applications

In probability theory, the theory of large deviations concerns the asymptotic be-haviour of remote tails of sequences of probability distributions. It may be effectively applied to gather information out of a probabilistic model. Thus, theory of large devi-ations finds its applications in mathematical finance and risk management.In 1970’s, Freidhin and Wentzell derived both the sample path diffusion large de-viation principle and the asymptotics of the problem of exit from a domain. After that, the theory of small perturbation large deviations for stochastic differential equations has been extensively developed.Like the large deviations, the moderate deviation problems arise in the theory of statistical inference quite naturally. The moderate deviations approximation bridge the gap between a central limit approximation and a large deviations approximation. The quadratic form of the moderate deviation principle’s rate function allows for the explicit minimization and in particular, it allows to obtain an asymptotic evaluation for the exit time.In this thesis, we study the Freidhin-Wentzell’s type moderate deviation principle for several stochastic dynamics, including the positive diffusions, stochastic Volterra E-quation, stochastic heat equation with spatially correlated noise and fractional stochas-tic heat equation with spatially correlated noise.