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We study stability, long-time behavior and moment estimates for stochasticevolution equations with additive Wiener noise and with singular drift given bya divergence type quasilinear diffusion operator which may not necessarilyexhibit a homogeneous diffusivity. Our results cover the singular stochastic$p$-Laplace equations and, more generally, singular stochastic $\Phi$-Laplaceequations with zero Dirichlet boundary conditions. We obtain improved momentestimates and quantitative convergence rates of the ergodic semigroup to theunique invariant measure, classified in a systematic way according to thedegree of local degeneracy of the potential at the origin. We obtain newconcentration results for the invariant measure and establish maximaldissipativity of the associated Kolmogorov operator. In particular, we recoverthe results for the curve shortening flow in the plane by Es-Sarhir, vonRenesse and Stannat, NoDEA 16(9), 2012, and improve the results by Liu andT\"olle, ECP 16, 2011.

Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation