We study models of the motion by mean curvature of an (1+1)-dimensional interface with random forcing. For the well-posedness we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution equations in the variational framework of Krylov-Rozovskiȋ, replacing the standard coercivity assumption by a Lyapunov-type condition. We also study the long-term behavior, showing that the homogeneous normal noise model [N. Dirr, S. Luckhaus, and M. Novaga, periodic boundary conditions converges to a spatially constant profile whose height behaves like a Brownian motion. For the additive vertical noise model with Dirichlet boundary conditions we show ergodicity, using the lower bound technique for Markov semigroups by Komorowski, Peszat and Szarek [Ann.

We use a coupling method for functional stochastic differential equations with bounded memory to establish an analogue of Wang's dimension-free Harnack inequality [13]. The strong Feller property for the corresponding segment process is also obtained.

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Abstract. We establish moment estimates for the invariant measure μ of a stochastic partial differential equation describing motion by mean curvature flow in (1+1) dimension, leading to polynomial stability of the associated Markov semigroup. We also prove maximal dissipativity on L 1 (μ) for the related Kolmogorov operator. Mathematics Subject Classification (2000). 47D07, 60H15, 35R60.

We study existence and a priori estimates of invariant measures µ for SPDE with local Lipschitz drift coefficients. Furthermore, we discuss the corresponding parabolic Cauchy-problem in L 1 (µ). Particular emphasis will be put on stochastic reaction diffusion equations.