Self consistent transfer operators. Invariant measures, convergence to equilibrium, linear response and control of the statistical properties

TL;DR

This paper develops a comprehensive framework for self consistent transfer operators, analyzing their invariant measures, convergence, stability, and response, applicable to various coupled dynamical systems with potential control over their statistical behavior.

Contribution

It introduces a general approach to self consistent transfer operators, proving existence, convergence, stability, and response results, extending beyond weak coupling regimes and applying to diverse systems.

Findings

Existence of invariant measures under broad conditions

Quantitative results on convergence speed to equilibrium

Methods for controlling statistical properties via optimal coupling

Abstract

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented in the paper are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We apply the general statements to examples of different nature: coupled continuous maps, coupled expanding maps, coupled systems with additive noise, systems made of \emph{different maps }coupled by a mean field interaction and other examples of self consistent transfer operators not coming from coupled maps. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.