  In this paper, the dimension-free Harnack inequality is proved for the
associated transition semigroups to a large class of stochastic evolution
equations with monotone drifts. As applications, the ergodicity, hyper-(or
ultra-)contractivity and compactness are established for the corresponding
transition semigroups. Moreover, the exponential convergence of the transition
semigroups to invariant measure and the existence of a spectral gap are also
derived. The main results are applied to many concrete stochastic evolution
equations such as stochastic reaction-diffusion equations, stochastic porous
media equations and the stochastic p-Laplace equation in Hilbert space.
