Weak Convergence Approach For The Large Deviation Principle Of Slow-fast Stochastic Partial Differential Equations

Large deviation principle mainly describes the occurrence mechanism of rare events in complex systems.This thesis aims to prove the large deviation principle of a class of coupled two-time-scale stochastic partial differential equations(SPDE).Freidlin and Wentzell[14]derived large deviation estimates for a class of diffusion process with small perturbations,and the asymptotic theory of the escape problem,requiring complicated and sophisticated calculations.Recently,Dupuis and Ellis[9]invented a weak convergence ap-proach,which build the large deviation principle simply and effectively.By using varia-tional formulas Laplace transform of the bounded continuous function,then obtain the large deviation principle by proving the Laplace principle which is equivalent to large deviation principle on Polish space.This method requires only some of the bounded estimations of the solution and the convergence of the solution,avoids some relatively complex exponential estimations,which reduces reduces the difficulty of proving the large deviation principle.so far,there is few results on large deviations by weak convergence for slow-fast stochastic partial differential equation due to the fact that a drift transformation leads the fast system to become a non-autonomous system,thus it is impossible to estimate the large deviation by averaging the slow system.In this paper,we mainly study a simple case of two-time-scale s-tochastic partial differential equation,where the fast equation in the model is linear.Then,by solving the slow-fast equation and substituting the solution of the fast equation into the slow equation,the large deviation principle is proved by using the weak convergence approach.