ERGODICITY OF STOCHASTIC CURVE SHORTENING FLOW IN THE PLANE

作者：Es Sarhir Abdelhadi^*; von Renesse Max K

来源：SIAM Journal on Mathematical Analysis, 2012, 44(1): 224-244.

DOI：10.1137/100798235

• degenerate stochastic equations
• monotonicity
• e-property

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-Rozovskii, 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. Poincare Anal. Non Lineaire, 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].

• 出版日期2012