Open AccessQuadratic response and speed of convergence of invariant measures in the zero-noise limit
Author(s) -
Stefano Galatolo,
Hugo Marsan
Publication year - 2021
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2021078
Subject(s) - mathematics , piecewise , limit (mathematics) , quadratic equation , zero (linguistics) , noise (video) , invariant measure , white noise , convergence (economics) , invariant (physics) , mathematical analysis , computer science , statistics , geometry , mathematical physics , linguistics , philosophy , artificial intelligence , economics , image (mathematics) , ergodic theory , economic growth
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.