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 and Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223
References:
[1]

P. Billingsley, Convergence of Probability Measures, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, Translated from the Russian by A. B. Sosinskiĭ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.

[3]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third edition, translated from the 1979 Russian original by Joseph Szücs, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.

[4]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091. doi: 10.1214/08-AOP392.

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, Corrected reprint of the 1985 original, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274, Springer-Verlag, Berlin, 1994.

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Corrected reprint of the 1985 original, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 275, Springer-Verlag, Berlin, 1994.

[7]

J. I. Kifer, Some theorems on small random perturbations of dynamical systems, Uspehi Mat. Nauk, 29 (1974), 205-206.

[8]

S. B. Kuksin and A. L. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation, J. Math. Pures Appl. (9), 89 (2008), 400-428. doi: 10.1016/j.matpur.2007.12.003.

[9]

S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Statist. Phys., 115 (2004), 469-492. doi: 10.1023/B:JOSS.0000019830.64243.a2.

[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), 134-142. doi: 10.1134/S0081543807040098.

[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), 134-142. doi: 10.1134/S0081543807040098.

[12]

S. B. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions, Geom. Funct. Anal., 20 (2010), 1431-1463. doi: 10.1007/s00039-010-0103-6.

[13]

S. B. Kuksin, Weakly nonlinear stochastic CGL equations, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1033-1056.

[14]

E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[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), 123-144. doi: 10.1111/1467-9590.01427.

[16]

N. I. Portenko, Generalized Diffusion Processes, Translated from the 1982 Russian original by H. H. McFaden, Translations of Mathematical Monographs, 83, American Mathematical Society, Providence, Rhode Island, 1990.

[17]

D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981/82), 137-151. doi: 10.1007/BF01206949.

[18]

Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89.

[19]

Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.

[20]

D. W. Stroock, Partial Differential Equations for Probabilists, Cambridge Studies in Advanced Mathematics, 112, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511755255.

[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, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, 333-359.

[22]

D. W. Stroock and S. R. Srinivasa Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233, Springer-Verlag, Berlin, 1979.

[23]

S. R. S. Varadhan, Stochastic Processes, Courant Lecture Notes in Mathematics, 16, Courant Institute of Mathematical Sciences, New York, 2007.

[24]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

show all references

References:
[1]

P. Billingsley, Convergence of Probability Measures, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, Translated from the Russian by A. B. Sosinskiĭ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.

[3]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third edition, translated from the 1979 Russian original by Joseph Szücs, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.

[4]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091. doi: 10.1214/08-AOP392.

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, Corrected reprint of the 1985 original, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274, Springer-Verlag, Berlin, 1994.

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Corrected reprint of the 1985 original, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 275, Springer-Verlag, Berlin, 1994.

[7]

J. I. Kifer, Some theorems on small random perturbations of dynamical systems, Uspehi Mat. Nauk, 29 (1974), 205-206.

[8]

S. B. Kuksin and A. L. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation, J. Math. Pures Appl. (9), 89 (2008), 400-428. doi: 10.1016/j.matpur.2007.12.003.

[9]

S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Statist. Phys., 115 (2004), 469-492. doi: 10.1023/B:JOSS.0000019830.64243.a2.

[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), 134-142. doi: 10.1134/S0081543807040098.

[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), 134-142. doi: 10.1134/S0081543807040098.

[12]

S. B. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions, Geom. Funct. Anal., 20 (2010), 1431-1463. doi: 10.1007/s00039-010-0103-6.

[13]

S. B. Kuksin, Weakly nonlinear stochastic CGL equations, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1033-1056.

[14]

E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[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), 123-144. doi: 10.1111/1467-9590.01427.

[16]

N. I. Portenko, Generalized Diffusion Processes, Translated from the 1982 Russian original by H. H. McFaden, Translations of Mathematical Monographs, 83, American Mathematical Society, Providence, Rhode Island, 1990.

[17]

D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981/82), 137-151. doi: 10.1007/BF01206949.

[18]

Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89.

[19]

Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.

[20]

D. W. Stroock, Partial Differential Equations for Probabilists, Cambridge Studies in Advanced Mathematics, 112, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511755255.

[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, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, 333-359.

[22]

D. W. Stroock and S. R. Srinivasa Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233, Springer-Verlag, Berlin, 1979.

[23]

S. R. S. Varadhan, Stochastic Processes, Courant Lecture Notes in Mathematics, 16, Courant Institute of Mathematical Sciences, New York, 2007.

[24]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

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