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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

Stefano Galatolo 1, and  Hugo Marsan 2,

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.

Keywords: Zero noise limit, random dynamical system, linear response, statistical stability.
Mathematics Subject Classification: Primary: 37C40; Secondary: 37H30, 37C30, 37E10.
Citation: Stefano Galatolo, Hugo Marsan. Quadratic response and speed of convergence of invariant measures in the zero-noise limit. Discrete and 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. 
[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.

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