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

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.

Non-hyperbolicity at large scales of a high-dimensional chaotic system

“…These limit systems may exhibit non-trivial and sometimes complex dynamics [30,31]. With certain smooth hyperbolic subsystems and sufficiently weak couplings, linear responses have been proven to exist in thermodynamic limit systems [32,33]. On the other hand, [12] presented a mean-field coupled system whose thermodynamic limit’s response to perturbations appeared to be non-smooth.…”