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Ergodicity and invariant measure approximation of the stochastic Cahn-Hilliard equation via an explicit fully discrete scheme

Nan Deng, Yibo Wang, Wanrong Cao

TL;DR

This work advances the long-time analysis of the stochastic Cahn–Hilliard equation with additive space–time white noise by proving a unique invariant measure on the mass-conserving space $H_eta$, while showing non-uniqueness on $L^2$. It then introduces an explicit fully discrete scheme—finite differences in space with a strongly tamed exponential Euler time integrator—and establishes uniform-in-time moment bounds and convergence with explicit rates. A novel mass-preserving minorization framework for Neumann boundary conditions yields geometric ergodicity and quantitative convergence of numerical invariant measures, together with strong laws of large numbers for both the continuous and discrete dynamics. Numerical experiments corroborate the theory, confirming long-time convergence, invariant-measure approximation accuracy, and practical efficiency of the explicit scheme for ergodic estimation.

Abstract

This paper investigates the stochastic Cahn-Hilliard equation (SCHE) driven by additive space-time white noise. We first refine the analytical ergodic theory by proving that the continuum equation admits a unique invariant measure in the more regular state space H_α, extending the classical result of Da Prato and Debussche (1996) on the negative Sobolev space $\dot{H}^{-1}_α$. To approximate long-time behaviour, we introduce an explicit fully discrete scheme that combines a finite-difference spatial discretization with a strongly tamed exponential Euler method in time. Uniform-in-time moment bounds in the $L^\infty$-norm are established for the numerical solution, and a uniform strong convergence estimate with an explicit rate is derived for the fully discrete approximation. Exploiting a mass-preserving minorization tailored to Neumann boundary conditions, we further show that the numerical scheme is geometrically ergodic and possesses a unique invariant measure, together with polynomial-order error bounds for approximating the exact invariant measure. Strong laws of large numbers are proved for both the continuous and discrete systems, ensuring almost-sure convergence of temporal averages to the corresponding ergodic limits. Numerical experiments corroborate the theoretical findings, including the long-time strong convergence and the accuracy of invariant measure approximation. Overall, the results provide a complete analytical and numerical framework for investigating the long-time statistical behaviour of the SCHE.

Ergodicity and invariant measure approximation of the stochastic Cahn-Hilliard equation via an explicit fully discrete scheme

TL;DR

This work advances the long-time analysis of the stochastic Cahn–Hilliard equation with additive space–time white noise by proving a unique invariant measure on the mass-conserving space , while showing non-uniqueness on . It then introduces an explicit fully discrete scheme—finite differences in space with a strongly tamed exponential Euler time integrator—and establishes uniform-in-time moment bounds and convergence with explicit rates. A novel mass-preserving minorization framework for Neumann boundary conditions yields geometric ergodicity and quantitative convergence of numerical invariant measures, together with strong laws of large numbers for both the continuous and discrete dynamics. Numerical experiments corroborate the theory, confirming long-time convergence, invariant-measure approximation accuracy, and practical efficiency of the explicit scheme for ergodic estimation.

Abstract

This paper investigates the stochastic Cahn-Hilliard equation (SCHE) driven by additive space-time white noise. We first refine the analytical ergodic theory by proving that the continuum equation admits a unique invariant measure in the more regular state space H_α, extending the classical result of Da Prato and Debussche (1996) on the negative Sobolev space . To approximate long-time behaviour, we introduce an explicit fully discrete scheme that combines a finite-difference spatial discretization with a strongly tamed exponential Euler method in time. Uniform-in-time moment bounds in the -norm are established for the numerical solution, and a uniform strong convergence estimate with an explicit rate is derived for the fully discrete approximation. Exploiting a mass-preserving minorization tailored to Neumann boundary conditions, we further show that the numerical scheme is geometrically ergodic and possesses a unique invariant measure, together with polynomial-order error bounds for approximating the exact invariant measure. Strong laws of large numbers are proved for both the continuous and discrete systems, ensuring almost-sure convergence of temporal averages to the corresponding ergodic limits. Numerical experiments corroborate the theoretical findings, including the long-time strong convergence and the accuracy of invariant measure approximation. Overall, the results provide a complete analytical and numerical framework for investigating the long-time statistical behaviour of the SCHE.

Paper Structure

This paper contains 13 sections, 24 theorems, 232 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant measure for the stochastic Cahn--Hilliard equation
  4. Key estimates for two auxiliary equations
  5. Existence and uniqueness of invariant measure
  6. Fully discrete scheme and uniform strong convergence
  7. Uniform moment boundedness of the numerical solution
  8. Uniform strong convergence of the numerical scheme
  9. Approximation of invariant measures
  10. Existence and uniqueness of invariant measure for the numerical solution
  11. Approximation of invariant measure
  12. Strong law of large numbers
  13. Numerical example

Key Result

Lemma 2.1

For any $\gamma, \gamma_1, \gamma_2 \in[0, \infty),\ \gamma_3 \in[0,1)$ satisfying $\gamma_1-\gamma_2+2\gamma_3 >0$ , and $\hat{u} \in H^{1} \cap \dot{H}$, the following inequalities hold: where $g:[0,\infty)\to\mathbb{R}$ is defined by $g(0)=1$ and $g(x)=0$ for all $x>0$, and $C>0$ is a constant independent of $s$, $t$, and $\hat{u}$.

Figures (3)

  • Figure 6.1: The time average of $\mathbb{E}\left[\phi(u^{\tau,N}(u_0;t,\cdot))\right]$ started from different initial values.
  • Figure 6.2: The time average of $\phi(u^{\tau,N}(u_0;t,\cdot))$ started from different initial values.
  • Figure 6.3: Ergodic limits for initial conditions in different spaces.

Theorems & Definitions (52)

  • Lemma 2.1
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 42 more