Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. Roughly speaking, besides the assumptions for existence and uniqueness of the solution, one only need assume some additional assumptions on diffusion coefficient in order to obtain Large deviation principle for the distribution of solution. As applications we can apply the main result to different type examples of SPDEs (e.g. stochastic reaction-diffusion equation, stochastic porous media and fast diffusion equations, stochastic p-Laplacian equation) in Hilbert space. The weak convergence approach is employed to verify the Laplace principle, which is equivalent to large deviation principle in our framework. AMS subject Classification: 60F10, 60H15.