3. Semilinear Stochastic Partial Differential Equations

In this chapter we establish Harnack/shift Harnack inequalities and derivative formulas for the semigroup associated with mild solutions of semilinear stochastic differential equations on Hilbert spaces. For simplicity, we consider only single-valued equations with a time-homogeneous linear operator; see Da Prato et al. (J. Funct. Anal. 257:992–1017, 2009), Ouyang (Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14:261–278, 2011), and Zhang (Probab. Lett. 83:1184–1192, 2013; Shift Harnack inequality and integration by part formula for semilinear SPDE, arXiv:1208.2425) for a study of equations possibly with a time-dependent linear operator and a multivalued nonlinear term. In the first section we introduce finite-dimensional approximations of mild solutions, which will then be used in the other sections to derive results from existing ones in finite dimensions. Materials in this chapter are modified from Wang (Ann. Probab. 39:1449–1467, 2011; Integration by parts formula and shift Harnack inequality for stochastic equations, arXiv:1203.4023), Wang and Zhang (Log-Harnack inequality for mild solutions of SPDEs with strongly multiplicative noise, arXiv:1210.6416).