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

Abstract

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 stochastic $p$-Laplace equations and, more generally, singular stochastic $\Phi$-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, and improve the results by Liu and Tölle, ECP 16, 2011.

24 pages, 58 references

References in corpus (6)

• SPDE in Hilbert Space with Locally Monotone Coefficients
• Stability of solutions to stochastic partial differential equations
• Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations
• Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise
• Kolmogorov Equations for Randomly Perturbed Generalized Newtonian Fluids
• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions