  In this paper, a new decay estimate for a class of stochastic evolution
equations with weakly dissipative drifts is established, which directly implies
the uniqueness of invariant measures for the corresponding transition
semigroups. Moreover, the existence of invariant measures and the convergence
rate of corresponding transition semigroup to the invariant measure are also
investigated. As applications, the main results are applied to singular
stochastic $p$-Laplace equations and stochastic fast diffusion equations, which
solves an open problem raised by Barbu and Da Prato in [Stoc. Proc. Appl.
120(2010), 1247-1266].
