Effective Dynamical System Of SPDEs Driven By Additive Noise

This thesis concerns the effective dynamical systems for stochastic partial differential equations with quadratic nonlinearities driven by degenerate additive noise.Through using stochastic analysis and multi-scale theory,the solution of the original equation is projected into the finite-dimensional kernel space,in order to obtain an effective dynamic system describing the evolution of the dominant modes.It is further proved that the new dynamic systems can effectively approximate the original stochastic partial differential equations,and the approximation form and convergence rate are given.More precisely,we consider two classes of stochastic partial differential equations.One is stochastic partial differential equations with singular perturbation in the noise term,and the other is stochastic partial differential equations with singular perturbation in the linear operator term.By constructing an appropriate time scale transformation,and using It(?) formula and Taylor formula as well as the stopping time technique to eliminate the influence of the nonkernel space elements,it is deduced the effective dynamic system containing only kernel space element.Finally,it is rigorously proved that the two classes of stochastic partial differential equations can be respectively approximated by the effective dynamical systems.