Large Deviation Principles For The Stochastic Heat Equation And Wave Equation With Rough Noise

The large deviation principle concerns the exponential decay of the probability measures of the cetain kinds of extreme or tail events.Freidlin and Wentzell discussed both the sample diffusion large deviation principle and asymptotic behaviour of the problem of exit from a domain.An important approach of investigating the large deviation principle for stochastic(partial)differential equations,especially stochastic partial differential equations driven by the multiplicative noise,is the well-known weak convergence method.This thesis is a study on the large deviation principle for several stochastic partial differential equations by the weak convergence method and a quadratic transportation cost inequality for the reflected stochastic partial differential equation.There are seven parts of the content of this thesis.In Chapter 1,we introduce the research background and the recent development of the large deviation principle and the quadratic transportation cost inequality of stochastic partial differential equations,and main results of this thesis are also introduced.In Chapter 2,we recall some standard definitions and results of the large deviation principle,and give the weak convergence criterion for the large deviation principle.In Chapter 3,we study the Freidlin-Wentzell large deviation principle for the one dimensional nonlinear stochastic heat equation driven by the rough noise by the weak convergence method.We remove the restriction σ(t,x,0)=0.We introduce the decay weight function and the weighted solution space,and use the approximation technique to prove the tightness and the convergence.In Chapter 4,we study the Freidlin-Wentzell large deviation principle for the one dimensional nonlinear stochastic wave equation driven by the rough noise by the weak convergence method.We use the technique that the Green function was decomposed to four complicated parts to overcome this difficult brought by a lack of semigroup property of the Green function of the wave equation.In Chapter 5,we study the Freidlin-Wentzell large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise by the weak convergence method.Here,we need to establish some precise estimates.In Chapter 6,we study the Freidlin-Wentzell large deviation principle for the reflected stochastic partial differential equation driven by the space-time white noise on the infinite spatial domain by the weak convergence method.As the domain of the spatial variable is infinite,we need introduce the weighted sup-normal space.In Chapter 7,for the reflected stochastic partial differential equation driven by the space-time white noise on the infinite spatial domain,we establish a quadratic transportation cost inequality under the weighted space.