November  2021, 41(11): 5303-5327. doi: 10.3934/dcds.2021078

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

Received  July 2020 Revised  February 2021 Published  November 2021 Early access  May 2021

Fund Project: The work is partially supported by the research project PRIN 2017S35EHN_004 "Regular and stochastic behavior in dynamical systems" of the Italian Ministry of Education and Research. The authors whish to thank Ecole Normale Paris Saclay and Università di Pisa for the organization of the international master stage "Stage d'initiation à la recherche M1" in which framework the work has been done. The authors also whish to thank W. Bahsoun and J. Sedro for useful discussions about zero noise limits and response

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 [25]). This is obtained by providing an explicit formula for the first and second term in the Taylor's expansion of the response of the stationary measure to the small noise perturbation. These terms depend on important features of the dynamics and of the noise which is perturbing it, as its average and variance.

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.

Citation: Stefano Galatolo, Hugo Marsan. Quadratic response and speed of convergence of invariant measures in the zero-noise limit. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5303-5327. doi: 10.3934/dcds.2021078
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[3]

J. F. Alves and V. Araújo, Random perturbations of nonuniformly expanding maps, Astérisque, 286 (2003), 25-62.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

W. BahsounS. GalatoloI. Nisoli and X. Niu, A rigorous computational approach to linear response, Nonlinearity, 31 (2018), 1073-1109.  doi: 10.1088/1361-6544/aa9a88.  Google Scholar

[8]

V. Baladi, Linear response, or else, Proceedings of the International Congress of Mathematicians–Seoul 2014, (2014), 525-545.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.   Google Scholar
[15]

S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, arXiv: 1510.02615 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. GalatoloM. Monge and I. Nisoli, Existence of noise induced order, a computer aided proof, Nonlinearity, 33 (2020), 4237-4276.  doi: 10.1088/1361-6544/ab86cd.  Google Scholar

[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.  Google Scholar

[21]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.  Google Scholar

[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.  Google Scholar

[23]

Yu. Kifer, On small random perturbations of some smooth dynamical systems, Math. USSR Ivestija, 8 (1974), 1083-1107.   Google Scholar

[24]

K. Krzyzewski, Some results on expanding mappings, Ast Risque, 50 (1977), 205-218.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[28]

R. J. Metzger, Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.  doi: 10.1007/s002200000220.  Google Scholar

[29]

M. Pollicott and P. Vytovna, Linear response and periodic points, Nonlinearity, 29 (2016), 3047-3066.  doi: 10.1088/0951-7715/29/10/3047.  Google Scholar

[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.  Google Scholar

[31]

J. Sedro, Regularity result for fixed points, with applications to linear response, Nonlinearity, 31 (2018), 1417-1440.  doi: 10.1088/1361-6544/aaa10b.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34] M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics 145, Cambridge University Press, 2014.  doi: 10.1017/CBO9781139976602.  Google Scholar
[35]

M. Viana, Stochastic Dynamics of Deterministic Systems, IMPA, Rio de Janeiro, 1997. Google Scholar

[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.  Google Scholar

[37]

L.-S. Young, Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems, 6 (1986), 311-319.  doi: 10.1017/S0143385700003473.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[3]

J. F. Alves and V. Araújo, Random perturbations of nonuniformly expanding maps, Astérisque, 286 (2003), 25-62.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

W. BahsounS. GalatoloI. Nisoli and X. Niu, A rigorous computational approach to linear response, Nonlinearity, 31 (2018), 1073-1109.  doi: 10.1088/1361-6544/aa9a88.  Google Scholar

[8]

V. Baladi, Linear response, or else, Proceedings of the International Congress of Mathematicians–Seoul 2014, (2014), 525-545.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.   Google Scholar
[15]

S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, arXiv: 1510.02615 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. GalatoloM. Monge and I. Nisoli, Existence of noise induced order, a computer aided proof, Nonlinearity, 33 (2020), 4237-4276.  doi: 10.1088/1361-6544/ab86cd.  Google Scholar

[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.  Google Scholar

[21]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.  Google Scholar

[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.  Google Scholar

[23]

Yu. Kifer, On small random perturbations of some smooth dynamical systems, Math. USSR Ivestija, 8 (1974), 1083-1107.   Google Scholar

[24]

K. Krzyzewski, Some results on expanding mappings, Ast Risque, 50 (1977), 205-218.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[28]

R. J. Metzger, Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.  doi: 10.1007/s002200000220.  Google Scholar

[29]

M. Pollicott and P. Vytovna, Linear response and periodic points, Nonlinearity, 29 (2016), 3047-3066.  doi: 10.1088/0951-7715/29/10/3047.  Google Scholar

[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.  Google Scholar

[31]

J. Sedro, Regularity result for fixed points, with applications to linear response, Nonlinearity, 31 (2018), 1417-1440.  doi: 10.1088/1361-6544/aaa10b.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34] M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics 145, Cambridge University Press, 2014.  doi: 10.1017/CBO9781139976602.  Google Scholar
[35]

M. Viana, Stochastic Dynamics of Deterministic Systems, IMPA, Rio de Janeiro, 1997. Google Scholar

[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.  Google Scholar

[37]

L.-S. Young, Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems, 6 (1986), 311-319.  doi: 10.1017/S0143385700003473.  Google Scholar

Figure 1.  Lipschitz approximation of a discontinuity, graphical representation of $ h_0 $ and $ f_a $ ($ a = 3 $)
Figure 2.  Lipschitz approximation of a discontinuity, rescaling of the problem
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