[Paper Review] Ergodicity of Stochastic Curve Shortening Flow in the Plane

Abdelhadi Es–Sarhir, M.-K. von Renesse|arXiv (Cornell University)|Mar 10, 2010

Stochastic processes and financial applications11 references7 citations

TL;DR

This paper establishes the well-posedness and ergodicity of a stochastic curve shortening flow in the plane driven by additive noise, using a variational framework for degenerate nonlinear SPDEs with a Lyapunov-type condition in lieu of coercivity. The key contribution is proving ergodicity via the e-property and a Lyapunov function with compact sublevel sets, extending the theory to non-coercive mean curvature-type SPDEs with additive noise.

ABSTRACT

We study a model of the motion by mean curvature of an (1+1) dimensional interface in a 2D Brownian velocity field. For the well-posedness of the model 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. Ergodicity is established for the case of additive noise, using the lower bound technique for Markov semigroups by Komorowski, Peszat and Szarek

Motivation & Objective

• To establish well-posedness for a degenerate stochastic curve shortening flow in the plane, which lacks standard coercivity.
• To develop a variational framework for nonlinear SPDEs that replaces coercivity with a Lyapunov-type condition.
• To prove ergodicity of the generalized solution under additive noise using the e-property and compact sublevel sets of a Lyapunov function.
• To extend the applicability of stochastic PDE theory to geometric flows with degenerate drift operators.

Proposed method

• Formulate the stochastic curve shortening flow as a nonlinear SPDE in the Gelfand triple $ H_0^1([0,1]) \subset L^2([0,1]) \subset H^{-1}([0,1]) $.
• Use the Krylov-Rozovskiï framework to prove existence and uniqueness under a Lyapunov-type condition instead of coercivity.
• Establish a generalized solution as a Markov process on $ L^2([0,1]) $, even for initial data in $ L^2 $, via approximation.
• Verify the e-property for the associated Feller semigroup using a Lyapunov function with compact sublevel sets.
• Apply the abstract ergodicity criterion from Komorowski, Peszat, and Szarek [6, Theorem 1] to prove ergodicity under additive noise.
• Use probabilistic estimates and Poincaré inequality to control the solution difference and ensure the lower bound condition for ergodicity.

Experimental results

Research questions

• RQ1 Can well-posedness be established for a stochastic curve shortening flow with degenerate drift and additive noise when standard coercivity fails?
• RQ2 Does the variational framework of Krylov-Rozovskiï admit a generalization to non-coercive SPDEs via a Lyapunov-type condition?
• RQ3 Is the generalized solution of the stochastic curve shortening flow ergodic under additive noise?
• RQ4 Can the e-property be verified for the Markov semigroup associated with this SPDE using a Lyapunov function with compact sublevel sets?
• RQ5 What is the long-time behavior of the solution process in $ L^2([0,1]) $, and does it converge to a unique invariant measure?

Key findings

• The SPDE (1.2) admits a unique generalized solution as a Markov process on $ L^2([0,1]) $, even for initial data in $ L^2 $, under the condition $ \sum_{i=1}^\infty (\text{Lip}(\phi_i))^2 \leq \Lambda^2 $.
• For initial data in $ H_0^1([0,1]) $, the solution is strong and uniquely defined via the variational method.
• The generalized solution satisfies the e-property and is ergodic under additive noise, as shown via the lower bound technique of Komorowski, Peszat, and Szarek.
• The existence of a Lyapunov function with compact sublevel sets enables the verification of the lower bound condition necessary for ergodicity.
• The transition probabilities satisfy $ \liminf_{T\to\infty} Q^T(x, B_\delta(0)) > 0 $ for all $ \delta > 0 $ and $ x \in L^2([0,1]) $, confirming uniform recurrence to neighborhoods of zero.
• The ergodicity result holds despite the deterministic flow not converging locally uniformly to equilibrium, which invalidates alternative criteria like [6, Theorem 3].