Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.11568 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917358565588992 |
|---|---|
| author | Chen, Huiping Chen, Yong Liu, Yong |
| author_facet | Chen, Huiping Chen, Yong Liu, Yong |
| contents | We investigate the global well-posedness and ergodicity of the complex Ginzburg-Landau equation with a general nonlinear term on the two-dimensional torus, driven by complex-valued space-time white noise. Due to the roughness of noise, the solution to this singular equation is a distribution-valued stochastic process. As a result, the nonlinear term is ill-defined and requires renormalization. We establish global well-posedness by combining the fixed point theorem with an estimate that decays over time. Moreover, we prove ergodicity by applying the Krylov-Bogoliubov theorem along with an asymptotic coupling argument. A crucial tool in our proof is the theory of complex multiple Wiener-Ito integrals, which enables direct estimates for random distributions themselves and provides a systematic framework for estimating complex Wick products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ergodicity for Ginzburg-Landau equation with complex-valued space-time white noise on two-dimensional torus Chen, Huiping Chen, Yong Liu, Yong Probability 60H17, 37A25 We investigate the global well-posedness and ergodicity of the complex Ginzburg-Landau equation with a general nonlinear term on the two-dimensional torus, driven by complex-valued space-time white noise. Due to the roughness of noise, the solution to this singular equation is a distribution-valued stochastic process. As a result, the nonlinear term is ill-defined and requires renormalization. We establish global well-posedness by combining the fixed point theorem with an estimate that decays over time. Moreover, we prove ergodicity by applying the Krylov-Bogoliubov theorem along with an asymptotic coupling argument. A crucial tool in our proof is the theory of complex multiple Wiener-Ito integrals, which enables direct estimates for random distributions themselves and provides a systematic framework for estimating complex Wick products. |
| title | Ergodicity for Ginzburg-Landau equation with complex-valued space-time white noise on two-dimensional torus |
| topic | Probability 60H17, 37A25 |
| url | https://arxiv.org/abs/2408.11568 |