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Volume 41, Issue 1
Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise

Zhihui Liu

DOI: 10.4208/cmr.2024-0042

Commun. Math. Res., 41 (2025), pp. 30-44.

Published online: 2025-03

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  • Abstract

We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments’ estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments’ estimate. Applying the main result to the stochastic Allen-Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.

  • Keywords

Numerical invariant measure, numerical ergodicity, stochastic Allen-Cahn equation.

  • AMS Subject Headings

Primary 60H35, Secondary 60H15, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-41-30, author = {Liu , Zhihui}, title = {Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise}, journal = {Communications in Mathematical Research }, year = {2025}, volume = {41}, number = {1}, pages = {30--44}, abstract = {

We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments’ estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments’ estimate. Applying the main result to the stochastic Allen-Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2024-0042}, url = {http://global-sci.org/intro/article_detail/cmr/23929.html} }
TY - JOUR T1 - Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise AU - Liu , Zhihui JO - Communications in Mathematical Research VL - 1 SP - 30 EP - 44 PY - 2025 DA - 2025/03 SN - 41 DO - http://doi.org/10.4208/cmr.2024-0042 UR - https://global-sci.org/intro/article_detail/cmr/23929.html KW - Numerical invariant measure, numerical ergodicity, stochastic Allen-Cahn equation. AB -

We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments’ estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments’ estimate. Applying the main result to the stochastic Allen-Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.

Liu , Zhihui. (2025). Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise. Communications in Mathematical Research . 41 (1). 30-44. doi:10.4208/cmr.2024-0042
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