2. Nonlinear Monotone Stochastic Partial Differential Equations

In the first section, we recall a general result concerning existence, uniqueness, and Itô’s formula for the norm square of solutions to nonlinear monotone stochastic differential equations in the framework of (Krylov and Rozovskii, Stochastic evolution equations, Plenum Publishing, 1981), which goes back to (Pardoux, C.R. Acad. Sci. 275:A101–A103, 1972) and (Pardoux, Equations aux dérivées partielles stochastiques non lineaires monotones: Etude de solutions fortes de type Ito, Thése Doct. Sci. Math. Univ. Paris Sud., 1975); then we apply this general result to the stochastic generalized porous media equations, the stochastic generalized fast-diffusion equations, and stochastic p-Laplacian equations. In the second and third sections, we establish the Harnack inequalities for a class of monotone stochastic differential equations with parameters α ≥ 1 and α ∈ (0, 1) respectively. Finally, the main results are illustrated by specific models in the last section. This chapter is organized according to (Liu, J. Evol. Equ. 9:747–770, 2009; Liu, Front. Math. China 6:449–472, 2011; Liu and Wang, J. Math. Anal. Appl. 342:651–662, 2008; Wang, Ann. Probab. 35:1333–1350, 2007).