Large Deviation Principle for Semilinear Stochastic Evolution Equations with Monotone Nonlinearity and Multiplicative Noise

We demonstrate the large deviation property for the mild solutions of stochastic evolution equations with monotone nonlinearity and multiplica- tive noise. This is achieved using the recently developed weak convergence method, in studying the large deviation principle. An It^o-type inequality is a main tool in the proofs. We also give two examples to illustrate the applications of the theorems.