October  2014, 34(10): 4223-4257. doi: 10.3934/dcds.2014.34.4223

Invariant measure selection by noise. An example

1. 

Mathematics Department and Department of Statisical Science, Statisical Science Duke University, Box 90320, Durham, NC 27708-0320, United States

2. 

Laboratoire d'Analyse, Topologie, Probabilités, Université de Provence 39, rue F. Joliot-Curie, F-13453 Marseille cedex 13, France

Received  March 2013 Revised  October 2013 Published  April 2014

We consider a deterministic system with two conserved quantities and infinity many invariant measures. However the systems possess a unique invariant measure when enough stochastic forcing and balancing dissipation are added. We then show that as the forcing and dissipation are removed a unique limit of the deterministic system is selected. The exact structure of the limiting measure depends on the specifics of the stochastic forcing.
Citation: Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223
References:
[1]

P. Billingsley, Convergence of Probability Measures,, Second edition, (1999). doi: 10.1002/9780470316962. Google Scholar

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory,, Translated from the Russian by A. B. Sosinskiĭ, (1982). doi: 10.1007/978-1-4615-6927-5. Google Scholar

[3]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, Third edition, (1979). doi: 10.1007/978-3-642-25847-3. Google Scholar

[4]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations,, Ann. Probab., 36 (2008), 2050. doi: 10.1214/08-AOP392. Google Scholar

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators,, Corrected reprint of the 1985 original, (1985). Google Scholar

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators,, Corrected reprint of the 1985 original, (1985). Google Scholar

[7]

J. I. Kifer, Some theorems on small random perturbations of dynamical systems,, Uspehi Mat. Nauk, 29 (1974), 205. Google Scholar

[8]

S. B. Kuksin and A. L. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation,, J. Math. Pures Appl. (9), 89 (2008), 400. doi: 10.1016/j.matpur.2007.12.003. Google Scholar

[9]

S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics,, J. Statist. Phys., 115 (2004), 469. doi: 10.1023/B:JOSS.0000019830.64243.a2. Google Scholar

[10]

S. B. Kuksin, Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model,, Tr. Mat. Inst. Steklova, 259 (2007), 134. doi: 10.1134/S0081543807040098. Google Scholar

[11]

S. B. Kuksin, Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model,, Tr. Mat. Inst. Steklova, 259 (2007), 134. doi: 10.1134/S0081543807040098. Google Scholar

[12]

S. B. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions,, Geom. Funct. Anal., 20 (2010), 1431. doi: 10.1007/s00039-010-0103-6. Google Scholar

[13]

S. B. Kuksin, Weakly nonlinear stochastic CGL equations,, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1033. Google Scholar

[14]

E. N. Lorenz, Deterministic nonperiodic flow,, Journal of the Atmospheric Sciences, 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Google Scholar

[15]

P. A. Milewski, E. G. Tabak and E. Vanden-Eijnden, Resonant wave interaction with random forcing and dissipation,, Studies in Applied Mathematics, 108 (2002), 123. doi: 10.1111/1467-9590.01427. Google Scholar

[16]

N. I. Portenko, Generalized Diffusion Processes,, Translated from the 1982 Russian original by H. H. McFaden, (1982). Google Scholar

[17]

D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors,, Comm. Math. Phys., 82 (): 137. doi: 10.1007/BF01206949. Google Scholar

[18]

Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms,, Funkcional. Anal. i Priložen, 2 (1968), 64. Google Scholar

[19]

Ja. G. Sinaĭ, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar

[20]

D. W. Stroock, Partial Differential Equations for Probabilists,, Cambridge Studies in Advanced Mathematics, (2008). doi: 10.1017/CBO9780511755255. Google Scholar

[21]

D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle,, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, (1970), 333. Google Scholar

[22]

D. W. Stroock and S. R. Srinivasa Varadhan, Multidimensional Diffusion Processes,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1979). Google Scholar

[23]

S. R. S. Varadhan, Stochastic Processes,, Courant Lecture Notes in Mathematics, (2007). Google Scholar

[24]

L.-S. Young, What are SRB measures, and which dynamical systems have them?,, J. Statist. Phys., 108 (2002), 733. doi: 10.1023/A:1019762724717. Google Scholar

show all references

References:
[1]

P. Billingsley, Convergence of Probability Measures,, Second edition, (1999). doi: 10.1002/9780470316962. Google Scholar

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory,, Translated from the Russian by A. B. Sosinskiĭ, (1982). doi: 10.1007/978-1-4615-6927-5. Google Scholar

[3]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, Third edition, (1979). doi: 10.1007/978-3-642-25847-3. Google Scholar

[4]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations,, Ann. Probab., 36 (2008), 2050. doi: 10.1214/08-AOP392. Google Scholar

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators,, Corrected reprint of the 1985 original, (1985). Google Scholar

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators,, Corrected reprint of the 1985 original, (1985). Google Scholar

[7]

J. I. Kifer, Some theorems on small random perturbations of dynamical systems,, Uspehi Mat. Nauk, 29 (1974), 205. Google Scholar

[8]

S. B. Kuksin and A. L. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation,, J. Math. Pures Appl. (9), 89 (2008), 400. doi: 10.1016/j.matpur.2007.12.003. Google Scholar

[9]

S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics,, J. Statist. Phys., 115 (2004), 469. doi: 10.1023/B:JOSS.0000019830.64243.a2. Google Scholar

[10]

S. B. Kuksin, Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model,, Tr. Mat. Inst. Steklova, 259 (2007), 134. doi: 10.1134/S0081543807040098. Google Scholar

[11]

S. B. Kuksin, Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model,, Tr. Mat. Inst. Steklova, 259 (2007), 134. doi: 10.1134/S0081543807040098. Google Scholar

[12]

S. B. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions,, Geom. Funct. Anal., 20 (2010), 1431. doi: 10.1007/s00039-010-0103-6. Google Scholar

[13]

S. B. Kuksin, Weakly nonlinear stochastic CGL equations,, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1033. Google Scholar

[14]

E. N. Lorenz, Deterministic nonperiodic flow,, Journal of the Atmospheric Sciences, 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Google Scholar

[15]

P. A. Milewski, E. G. Tabak and E. Vanden-Eijnden, Resonant wave interaction with random forcing and dissipation,, Studies in Applied Mathematics, 108 (2002), 123. doi: 10.1111/1467-9590.01427. Google Scholar

[16]

N. I. Portenko, Generalized Diffusion Processes,, Translated from the 1982 Russian original by H. H. McFaden, (1982). Google Scholar

[17]

D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors,, Comm. Math. Phys., 82 (): 137. doi: 10.1007/BF01206949. Google Scholar

[18]

Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms,, Funkcional. Anal. i Priložen, 2 (1968), 64. Google Scholar

[19]

Ja. G. Sinaĭ, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar

[20]

D. W. Stroock, Partial Differential Equations for Probabilists,, Cambridge Studies in Advanced Mathematics, (2008). doi: 10.1017/CBO9780511755255. Google Scholar

[21]

D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle,, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, (1970), 333. Google Scholar

[22]

D. W. Stroock and S. R. Srinivasa Varadhan, Multidimensional Diffusion Processes,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1979). Google Scholar

[23]

S. R. S. Varadhan, Stochastic Processes,, Courant Lecture Notes in Mathematics, (2007). Google Scholar

[24]

L.-S. Young, What are SRB measures, and which dynamical systems have them?,, J. Statist. Phys., 108 (2002), 733. doi: 10.1023/A:1019762724717. Google Scholar

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