| Abstract: |
| This work 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 on the more regular state space $H_\alpha$, extending the classical result of Da Prato & Debussche (1996) on the negative Sobolev space $\dot{H}^{-1}_\alpha$. 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. |
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