Stability of the invariant measure for the 3D stochastic cubic Ginzburg–Landau systems

Abstract

The current paper is devoted to 3D stochastic Ginzburg–Landau equations with degenerate random forcing. We establish the stability of stochastic systems by investigating the relationship between invariant measures under the action of transition semigroups corresponding to different sets of parameters. Towards this aim a new form of bound on the difference between solutions along with the spectral gap plays a significant role.

Keywords:

• Stochastic Ginzburg–Landau equations
• stochastic stability
• exponential ergodicity
• invariant measure

Mathematical Subject Classifications:

• 93E15
• 37A25
• 35R60

Acknowledgements

This paper was motivated mainly during a stay at the Mathematics Research Centre of the University of Warwick. The authors would like to thank Xue-mei Li, Martin Hairer and David Elworthy for their warm hospitality and useful discussions.

Disclosure statement

The authors declare no competing non-financial/financial interests.

Data availability statement

All is available.

Additional information

Funding

The manuscript is supported by the National Natural Science Foundation of China (NSF of China) [grant number 12071480].

Notes on contributors

Dengdi Chen

Dengdi Chen contributed to carrying out additional analyses and finalizing this paper.

Yan Zheng

Yan Zheng contributed the central idea and wrote the initial draft of the paper.