Continuity properties of transport coefficients in simple maps

TL;DR

This paper rigorously analyzes the continuity of transport coefficients, such as drift and diffusion, in families of simple maps with exponential mixing, providing both theoretical proofs and numerical verification of their regularity properties.

Contribution

The paper establishes rigorous continuity properties of transport coefficients in expanding interval maps, including precise modulus of continuity estimates and numerical validation.

Findings

D(l) is Lipschitz continuous up to quadratic logarithmic corrections.

Numerical verification confirms analytical continuity estimates.

Strong local variations in continuity properties are observed numerically.

Abstract

We consider families of dynamics that can be described in terms of Perron-Frobenius operators with exponential mixing properties. For piecewise C^2 expanding interval maps we rigorously prove continuity properties of the drift J(l) and of the diffusion coefficient D(l) under parameter variation. Our main result is that D(l) has a modulus of continuity of order O(|dl||log|dl|)^2), i.e. D(l) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are verified numerically for the latter class of maps by using exact formulas for the transport coefficients. We numerically observe strong local variations of all continuity properties.