
-
Previous Article
A multiplicity result for orthogonal geodesic chords in Finsler disks
- DCDS Home
- This Issue
-
Next Article
A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing
Quadratic response and speed of convergence of invariant measures in the zero-noise limit
1. | Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy, http://pagine.dm.unipi.it/a080288/ |
2. | Bâtiment Nord - 2S18. 4, avenue des Sciences, 91190 Gif-sur-Yvette, France |
We study the stochastic stability in the zero-noise limit from a quantitative point of view.
We consider smooth expanding maps of the circle perturbed by additive noise. We show that in this case the zero-noise limit has a quadratic speed of convergence, as suggested by numerical experiments and heuristics published by Lin, in 2005 (see [
We also consider the zero-noise limit from a quantitative point of view for piecewise expanding maps showing estimates for the speed of convergence in this case.
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
![]() ![]() |
[2] |
J. F. Alves,
Strong statistical stability of non-uniformly expanding maps, Nonlinearity, 17 (2004), 1193-1215.
doi: 10.1088/0951-7715/17/4/004. |
[3] |
J. F. Alves and V. Araújo,
Random perturbations of nonuniformly expanding maps, Astérisque, 286 (2003), 25-62.
|
[4] |
J. F. Alves and M. A. Khan,
Statistical instability for contracting Lorenz flows, Nonlinearity, 32 (2019), 4413-4444.
doi: 10.1088/1361-6544/ab2f48. |
[5] |
J. F. Alves and H. Vilarinho,
Strong stochastic stability for non-uniformly expanding maps, Ergodic Theory Dynam. Systems, 33 (2013), 647-692.
doi: 10.1017/S0143385712000077. |
[6] |
V. Araújo and A. Tahzibi,
Stochastic stability at the boundary of expanding maps, Nonlinearity, 18 (2005), 939-958.
doi: 10.1088/0951-7715/18/3/001. |
[7] |
W. Bahsoun, S. Galatolo, I. Nisoli and X. Niu,
A rigorous computational approach to linear response, Nonlinearity, 31 (2018), 1073-1109.
doi: 10.1088/1361-6544/aa9a88. |
[8] |
V. Baladi,
Linear response, or else, Proceedings of the International Congress of Mathematicians–Seoul 2014, Ⅲ (2014), 525-545.
|
[9] |
V. Baladi and M. Viana,
Strong stochastic stability and rate of mixing for unimodal maps, Ann. Sci. École Norm. Sup., 29 (1996), 483-517.
doi: 10.24033/asens.1745. |
[10] |
V. Baladi and L.-S. Young,
On the spectra of randomly perturbed expanding maps, Commun. Math. Phys., 156 (1993), 355-385.
doi: 10.1007/BF02098487. |
[11] |
M. Blank and G. Keller,
Stochastic stability versus localization in one-dimensional chaotic dynamical systems, Nonlinearity, 10 (1997), 81-107.
doi: 10.1088/0951-7715/10/1/006. |
[12] |
M. Blank and G. Keller,
Random perturbations of chaotic dynamical systems: Stability of the spectrum, Nonlinearity, 11 (1998), 1351-1364.
doi: 10.1088/0951-7715/11/5/010. |
[13] |
J.-P. Eckmann and D. Ruelle,
Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.
doi: 10.1103/RevModPhys.57.617. |
[14] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.
![]() ![]() |
[15] |
S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, arXiv: 1510.02615 |
[16] |
S. Galatolo, Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products, J. Éc. Polytech. Math., 5 (2018), 377–405.
doi: 10.5802/jep.73. |
[17] |
S. Galatolo,
Quantitative statistical stability and convergence to equilibrium. An application to maps with indifferent fixed points, Chaos Solitons Fractals, 103 (2017), 596-601.
doi: 10.1016/j.chaos.2017.07.005. |
[18] |
S. Galatolo and R. Lucena,
Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps,, Discrete Contin. Dyn. Syst., 40 (2020), 1309-1360.
doi: 10.3934/dcds.2020079. |
[19] |
S. Galatolo, M. Monge and I. Nisoli,
Existence of noise induced order, a computer aided proof, Nonlinearity, 33 (2020), 4237-4276.
doi: 10.1088/1361-6544/ab86cd. |
[20] |
S. Galatolo and J. Sedro, Quadratic response of random and deterministic dynamical systems, Chaos, 30 (2020), 023113, 15 pp.
doi: 10.1063/1.5122658. |
[21] |
S. Gouëzel and C. Liverani,
Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.
doi: 10.1017/S0143385705000374. |
[22] |
M. Jézéquel,
Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response, Discrete Contin. Dyn. Syst., 39 (2019), 927-958.
doi: 10.3934/dcds.2019039. |
[23] |
Yu. Kifer,
On small random perturbations of some smooth dynamical systems, Math. USSR Ivestija, 8 (1974), 1083-1107.
|
[24] |
K. Krzyzewski,
Some results on expanding mappings, Ast Risque, 50 (1977), 205-218.
|
[25] |
K. K. Lin,
Convergence of invariant densities in the small-noise limit, Nonlinearity, 18 (2005), 659-683.
doi: 10.1088/0951-7715/18/2/011. |
[26] |
C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical Systems, Part Ⅱ, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2003), 185–237. |
[27] |
C. Liverani, B. Saussol and S. Vaienti,
A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.
doi: 10.1017/S0143385799133856. |
[28] |
R. J. Metzger,
Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.
doi: 10.1007/s002200000220. |
[29] |
M. Pollicott and P. Vytovna,
Linear response and periodic points, Nonlinearity, 29 (2016), 3047-3066.
doi: 10.1088/0951-7715/29/10/3047. |
[30] |
D. Ruelle,
Nonequilibrium statistical mechanics near equilibrium: Computing higher-order terms, Nonlinearity, 11 (1998), 5-18.
doi: 10.1088/0951-7715/11/1/002. |
[31] |
J. Sedro,
Regularity result for fixed points, with applications to linear response, Nonlinearity, 31 (2018), 1417-1440.
doi: 10.1088/1361-6544/aaa10b. |
[32] |
W. Shen,
On stochastic stability of non-uniformly expanding interval maps, Proc. Lond. Math. Soc., 107 (2013), 1091-1134.
doi: 10.1112/plms/pdt013. |
[33] |
W. Shen and S. van Strien,
On stochastic stability of expanding circle maps with neutral fixed points, Dyn. Syst., 28 (2013), 423-452.
doi: 10.1080/14689367.2013.806733. |
[34] |
M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics 145, Cambridge University Press, 2014.
doi: 10.1017/CBO9781139976602.![]() ![]() ![]() |
[35] |
M. Viana, Stochastic Dynamics of Deterministic Systems, IMPA, Rio de Janeiro, 1997. |
[36] |
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. |
[37] |
L.-S. Young,
Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems, 6 (1986), 311-319.
doi: 10.1017/S0143385700003473. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
![]() ![]() |
[2] |
J. F. Alves,
Strong statistical stability of non-uniformly expanding maps, Nonlinearity, 17 (2004), 1193-1215.
doi: 10.1088/0951-7715/17/4/004. |
[3] |
J. F. Alves and V. Araújo,
Random perturbations of nonuniformly expanding maps, Astérisque, 286 (2003), 25-62.
|
[4] |
J. F. Alves and M. A. Khan,
Statistical instability for contracting Lorenz flows, Nonlinearity, 32 (2019), 4413-4444.
doi: 10.1088/1361-6544/ab2f48. |
[5] |
J. F. Alves and H. Vilarinho,
Strong stochastic stability for non-uniformly expanding maps, Ergodic Theory Dynam. Systems, 33 (2013), 647-692.
doi: 10.1017/S0143385712000077. |
[6] |
V. Araújo and A. Tahzibi,
Stochastic stability at the boundary of expanding maps, Nonlinearity, 18 (2005), 939-958.
doi: 10.1088/0951-7715/18/3/001. |
[7] |
W. Bahsoun, S. Galatolo, I. Nisoli and X. Niu,
A rigorous computational approach to linear response, Nonlinearity, 31 (2018), 1073-1109.
doi: 10.1088/1361-6544/aa9a88. |
[8] |
V. Baladi,
Linear response, or else, Proceedings of the International Congress of Mathematicians–Seoul 2014, Ⅲ (2014), 525-545.
|
[9] |
V. Baladi and M. Viana,
Strong stochastic stability and rate of mixing for unimodal maps, Ann. Sci. École Norm. Sup., 29 (1996), 483-517.
doi: 10.24033/asens.1745. |
[10] |
V. Baladi and L.-S. Young,
On the spectra of randomly perturbed expanding maps, Commun. Math. Phys., 156 (1993), 355-385.
doi: 10.1007/BF02098487. |
[11] |
M. Blank and G. Keller,
Stochastic stability versus localization in one-dimensional chaotic dynamical systems, Nonlinearity, 10 (1997), 81-107.
doi: 10.1088/0951-7715/10/1/006. |
[12] |
M. Blank and G. Keller,
Random perturbations of chaotic dynamical systems: Stability of the spectrum, Nonlinearity, 11 (1998), 1351-1364.
doi: 10.1088/0951-7715/11/5/010. |
[13] |
J.-P. Eckmann and D. Ruelle,
Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.
doi: 10.1103/RevModPhys.57.617. |
[14] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.
![]() ![]() |
[15] |
S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, arXiv: 1510.02615 |
[16] |
S. Galatolo, Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products, J. Éc. Polytech. Math., 5 (2018), 377–405.
doi: 10.5802/jep.73. |
[17] |
S. Galatolo,
Quantitative statistical stability and convergence to equilibrium. An application to maps with indifferent fixed points, Chaos Solitons Fractals, 103 (2017), 596-601.
doi: 10.1016/j.chaos.2017.07.005. |
[18] |
S. Galatolo and R. Lucena,
Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps,, Discrete Contin. Dyn. Syst., 40 (2020), 1309-1360.
doi: 10.3934/dcds.2020079. |
[19] |
S. Galatolo, M. Monge and I. Nisoli,
Existence of noise induced order, a computer aided proof, Nonlinearity, 33 (2020), 4237-4276.
doi: 10.1088/1361-6544/ab86cd. |
[20] |
S. Galatolo and J. Sedro, Quadratic response of random and deterministic dynamical systems, Chaos, 30 (2020), 023113, 15 pp.
doi: 10.1063/1.5122658. |
[21] |
S. Gouëzel and C. Liverani,
Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.
doi: 10.1017/S0143385705000374. |
[22] |
M. Jézéquel,
Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response, Discrete Contin. Dyn. Syst., 39 (2019), 927-958.
doi: 10.3934/dcds.2019039. |
[23] |
Yu. Kifer,
On small random perturbations of some smooth dynamical systems, Math. USSR Ivestija, 8 (1974), 1083-1107.
|
[24] |
K. Krzyzewski,
Some results on expanding mappings, Ast Risque, 50 (1977), 205-218.
|
[25] |
K. K. Lin,
Convergence of invariant densities in the small-noise limit, Nonlinearity, 18 (2005), 659-683.
doi: 10.1088/0951-7715/18/2/011. |
[26] |
C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical Systems, Part Ⅱ, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2003), 185–237. |
[27] |
C. Liverani, B. Saussol and S. Vaienti,
A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.
doi: 10.1017/S0143385799133856. |
[28] |
R. J. Metzger,
Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.
doi: 10.1007/s002200000220. |
[29] |
M. Pollicott and P. Vytovna,
Linear response and periodic points, Nonlinearity, 29 (2016), 3047-3066.
doi: 10.1088/0951-7715/29/10/3047. |
[30] |
D. Ruelle,
Nonequilibrium statistical mechanics near equilibrium: Computing higher-order terms, Nonlinearity, 11 (1998), 5-18.
doi: 10.1088/0951-7715/11/1/002. |
[31] |
J. Sedro,
Regularity result for fixed points, with applications to linear response, Nonlinearity, 31 (2018), 1417-1440.
doi: 10.1088/1361-6544/aaa10b. |
[32] |
W. Shen,
On stochastic stability of non-uniformly expanding interval maps, Proc. Lond. Math. Soc., 107 (2013), 1091-1134.
doi: 10.1112/plms/pdt013. |
[33] |
W. Shen and S. van Strien,
On stochastic stability of expanding circle maps with neutral fixed points, Dyn. Syst., 28 (2013), 423-452.
doi: 10.1080/14689367.2013.806733. |
[34] |
M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics 145, Cambridge University Press, 2014.
doi: 10.1017/CBO9781139976602.![]() ![]() ![]() |
[35] |
M. Viana, Stochastic Dynamics of Deterministic Systems, IMPA, Rio de Janeiro, 1997. |
[36] |
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. |
[37] |
L.-S. Young,
Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems, 6 (1986), 311-319.
doi: 10.1017/S0143385700003473. |


[1] |
Stefano Galatolo, Alfonso Sorrentino. Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 815-839. doi: 10.3934/dcds.2021138 |
[2] |
Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701 |
[3] |
Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002 |
[4] |
Shivam Dhama, Chetan D. Pahlajani. Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022018 |
[5] |
François Delarue, Franco Flandoli. The transition point in the zero noise limit for a 1D Peano example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4071-4083. doi: 10.3934/dcds.2014.34.4071 |
[6] |
Mika Yoshida, Kinji Fuchikami, Tatsuya Uezu. Realization of immune response features by dynamical system models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 531-552. doi: 10.3934/mbe.2007.4.531 |
[7] |
Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004 |
[8] |
Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757 |
[9] |
Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039 |
[10] |
Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477 |
[11] |
Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115 |
[12] |
Julian Newman. Synchronisation of almost all trajectories of a random dynamical system. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4163-4177. doi: 10.3934/dcds.2020176 |
[13] |
Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875 |
[14] |
Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161 |
[15] |
Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166 |
[16] |
Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122 |
[17] |
Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295 |
[18] |
Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure and Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038 |
[19] |
Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 |
[20] |
Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]