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

TL;DR

This paper investigates the rate at which invariant measures of smooth expanding maps on the circle converge as noise diminishes, revealing a quadratic speed of convergence and providing explicit formulas for the response to small noise perturbations.

Contribution

It establishes a quadratic convergence rate in the zero-noise limit for smooth expanding maps and derives explicit formulas for the response of stationary measures to small noise.

Findings

Zero-noise limit has quadratic speed of convergence.

Explicit formulas for first and second order response terms.

Estimates for convergence speed in piecewise expanding maps.

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 conjectured by Lin, in 2005, after numerical experiments (see arXiv:math/0406201 ). 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.