Quadratic response and speed of convergence of invariant measures in the zero-noise limit

Stefano Galatolo; Hugo Marsan

doi:10.3934/dcds.2021078

Article Contents

2021, Volume 41, Issue 11: 5303-5327. Doi: 10.3934/dcds.2021078

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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: June 30, 2020

Revised: January 31, 2021

Early access: May 2021

Published: November 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37C40; Secondary: 37H30, 37C30, 37E10.

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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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