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.
Similar content being viewed by others
Notes
Recall the definition of \(y^\pi (x_0, y_0)\) in Definition 2.1.
References
Arnold, V.I.: Sur la géométrie différentielle des groups de lie de dimension infinite et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier 16, 316–361 (1966)
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1978)
Arnold, V.I., Khesin, B.: Topological Methods in Hydrodynamics. Springer, Berlin (1998)
Berglund, N.: Kramers’ law: validity, derivations and generalizations. Markov Process. Relat. Fields 19, 459–490 (2013)
Bouchet, F., Morita, H.: Large-time behavior and asymptotic stability of the 2D Euler and linerized Euler equations. Phys. D 239, 948–966 (2010)
Bouchet, F., Sommeria, J.: Emergence of intense jets and Jupiter’s Great Red Spot as maximum-entropy structures. J. Fluid Mech. 464, 165–207 (2002)
Bouchet, F., Touchette, H.: Non-classical large deviations for a noisy system with non-isolated attractors. J. Stat. Mech. 2012, P05028 (2012)
Bouchet, F., Venallie, A.: Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227–295 (2012)
Dolgopyat, D., Koralov, L.: Averaging of Hamiltonian flows with an ergodic component. Ann. Probab. 36, 1999–2049 (2008)
Dolgopyat, D., Koralov, L.: Averaging of incompressible flows on two dimensional surfaces. J. Am. Math. Soc. 26(2), 427–449 (2013)
Dynkin, E.B.: One-dimensional continuous strong Markov processes. Theory Probab. Appl. IV(1), 1–52 (1959)
Elgindi, T., Hu, W., Šverák, V.: On 2d incompressible Euler equations with partial damping. Commun. Math. Phys. 355(1), 145–159 (2017)
Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (2005)
Feller, W.: Generalized second-order differential operators and their lateral conditions. Ill. J. Math. 1, 459–504 (1957)
Freidlin, M.: Sublimiting distributions and stabilization of solutions of parabolic equations with a small parameter. Sov. Math. Dokl. 235(5), 1042–1045 (1977)
Freidlin, M.: On stochastic perturbations of dynamical systems with a “rough” symmetry: hierarchy of Markov chains. J. Stat. Phys. 157(6), 1031–1045 (2014)
Freidlin, M., Hu, W.: On perturbations of the generalized Landau–Lifschitz dynamics. J. Stat. Phys. 144, 978–1008 (2011)
Freidlin, M., Hu, W.: On stochasticity in nearly-elastic systems. Stoch. Dyn. 12(3), 1150020 (2012)
Freidlin, M., Hu, W.: On second order elliptic equations with a small parameter. Commun. Partial Differ. Equ. 38(10), 1712–1736 (2013)
Freidlin, M., Hu, W., Wentzell, A.: Small mass asymptotic for the motion with vanishing friction. Stoch. Process. Appl. 123, 45–75 (2013)
Freidlin, M., Koralov, L.: Metastable distributions of Markov chains with rare transitions. J. Stat. Phys. 167(6), 1355–1375 (2017)
Freidlin, M., Koralov, L., Wentzell, A.: On Diffusions in Media with Pockets of Large Diffusivity. arXiv:1710.03555v1 [math.PR]
Freidlin, M., Koralov, L., Wentzell, A.: On the behavior of diffusion processes with traps. Ann. Probab. 45(5), 3202–3222 (2017)
Freidlin, M., Korlaov, L.: On stochastic perturbations of slowly changing dynamical systems. Nonlinearity 30(1), 445 (2016)
Freidlin, M., Wentzell, A.: On small random perturbations of dynamical systems. Russ. Math. Surv. 25(1), 1–56 (1970)
Freidlin, M., Wentzell, A.: Diffusion processes on graphs and the averaging principle. Ann. Probab. 21(4), 2215–2245 (1993)
Freidlin, M., Wentzell, A.: Random Perturbations of Hamiltonian Systems. Memoirs of the American Mathematical Society (1994)
Freidlin, M., Wentzell, A.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, Berlin (1998)
Freidlin, M., Wentzell, A.: On the Neumann problem for PDE’s with a small parameter and the corresponding diffusion processes. Probab. Theory Relat. Fields 152(1–2), 101–140 (2012)
Freidlin, M., Wentzell, A.: Random Perturbations of Dynamical Systems, 3rd edn. Springer, Berlin (2012)
Hu, W.: On metastability in nearly-elastic systems. Asymptot. Anal. 79(1–2), 65–86 (2012)
Hu, W., Šverák, V.: Dynamics of geodesic flows with random forcing on lie groups with left-invariant metrics. J. Nonlinear Sci. 28(6), 2249–2274 (2018)
Kuksin, S., Shirikyan, A.: Rigorous results in space-periodic two-dimensional turbulence. Phys. Fluids 29, 125106 (2017)
Mandl, P.: Analytical Treatment of One-Dimensional Markov Processes. Springer, Berlin (1968)
Martiosyan, D.: Large deviations for stationary measures of stochastic non-linear wave equations with smooth white noise. Commun. Pure Appl. Math. 70(9), 1754–1797 (2017)
Miller, J.: Statistical mechanics of Euler equations in two-dimensions. Phys. Rev. Lett. 65, 2137–2140 (1990)
Molchanov, S.A.: Martin boundary for invariant Markov processes on a solvable group. Theory Probab. Appl. 12, 310–314 (1967). (English translation)
Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207, 29–201 (2011)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)
Robert, R., Sommeria, J.: Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310 (1991)
Schneider, K., Farge, M.: Final states of decaying 2-d turbulence in bounded domains: influence of the geometry. Phys. D 237, 2228–2233 (2008)
Sommeria, J.: Two dimensional turbulence. In: New Trends Turbulence. Les Houches Summer School, New York: Springer, vol. 74, pp. 385–447 (2001)
Šverák, V.: Lecture notes of Selected Topics in Fluid Mechanics. University of Minnesota (2011–2012)
Tabling, P.: Two-dimensional turbulence, a physicist approach. Phys. Rep. 362(1), 1–62 (2002)
Tao, T.: The Euler–Arnold equation. https://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/. Accessed 8 Jan 2018
Willams, R.F.: The structure of Lorentz attractors. Publications Mathématiques de l’I.H.É.S tome 50, 73–99 (1979)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-019-00911-2
Keywords
- Random perturbations of dynamical system
- Group symmetry
- Invariant measure
- Nonlinear dynamics
- Irreversibility