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On the Long-Time Behavior of a Perturbed Conservative System with Degeneracy

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Abstract

We consider in this work a model conservative system subject to dissipation and Gaussian-type stochastic perturbations. The original conservative system possesses a continuous set of steady states and is thus degenerate. We characterize the long-time limit of our model system as the perturbation parameter tends to zero. The degeneracy in our model system carries features found in some partial differential equations related, for example, to turbulence problems.

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Notes

  1. Recall the definition of \(y^\pi (x_0, y_0)\) in Definition 2.1.

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Acknowledgements

The author would like to thank Professor Vladimír Šverák from University of Minnesota, USA for fruitful discussions on the formulation of his system (3) as the Euler–Arnold equation on the group of all affine transformations of a line, as well as its relation with fluid mechanics. He also would like to thank the anonymous referee, Professor Yong Liu from Peking University, Beijing, China and Professor Yong Ren from Anhui Normal University, Wuhu, Anhui, China for their valuable comments that improve the first version of this work.

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  1. Department of Mathematics and Statistics, Missouri University of Science and Technology (formerly University of Missouri, Rolla), Rolla, USA

    Wenqing Hu

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Hu, W. On the Long-Time Behavior of a Perturbed Conservative System with Degeneracy. J Theor Probab 33, 1266–1295 (2020). https://doi.org/10.1007/s10959-019-00911-2

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