Ergodicity of Stochastic Curve Shortening Flow in the Plane

Abstract

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–Rozovski, 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, Calc. Var. Partial Differential Equations, 13 (2001), pp. 405–425], [P. E. Souganidis and N. K. Yip, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 1–23] with 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. Probab., 38 (2010), pp. 1401–1443].