  We study stability, long-time behavior and moment estimates for stochastic
evolution equations with additive Wiener noise and with singular drift given by
a divergence type quasilinear diffusion operator which may not necessarily
exhibit a homogeneous diffusivity. Our results cover the singular $p$-Laplace
and, more generally, singular $\Phi$-Laplace equations with zero Dirichlet
boundary conditions. We obtain improved moment estimates and quantitative
convergence rates of the ergodic semigroup to the unique invariant measure,
classified in a systematic way according to the degree of local degeneracy of
the potential at the origin. We obtain new concentration results for the
invariant measure and establish maximal dissipativity of the associated
Kolmogorov operator. In particular, we recover the results for the curve
shortening flow in the plane by Es-Sarhir, von Renesse and Stannat, NoDEA
16(9), 2012.
