Ergodicity and invariant measure approximation of the stochastic Cahn-Hilliard equation via an explicit fully discrete scheme
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2025
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| author | Deng, Nan Wang, Yibo Cao, Wanrong |
| author_facet | Deng, Nan Wang, Yibo Cao, Wanrong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_01621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ergodicity and invariant measure approximation of the stochastic Cahn-Hilliard equation via an explicit fully discrete scheme Deng, Nan Wang, Yibo Cao, Wanrong Numerical Analysis 60H15, 65C30, 37A30, 60F15 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. |
| title | Ergodicity and invariant measure approximation of the stochastic Cahn-Hilliard equation via an explicit fully discrete scheme |
| topic | Numerical Analysis 60H15, 65C30, 37A30, 60F15 |
| url | https://arxiv.org/abs/2512.01621 |