Stability and approximation of random invariant densities for Lasota-Yorke map cocycles

TL;DR

This paper proves the stability of random invariant measures for Lasota-Yorke map cocycles under various perturbations, providing a theoretical foundation for numerical approximation methods like Fourier and Ulam's schemes.

Contribution

It introduces a general stability framework for random acims under broad perturbations, including convolution, finite-rank approximations, and static changes.

Findings

Stability of random acims under convolution perturbations

Validation of Fourier-based numerical approximation schemes

Effectiveness of Ulam's method for approximating random acims

Abstract

We establish stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations. Our family of random maps need not be close to a fixed map; thus, our results can handle very general driving mechanisms. We consider (i) perturbations via convolutions, (ii) perturbations arising from finite-rank transfer operator approximation schemes and (iii) static perturbations, perturbing to a nearby cocycle of Lasota-Yorke maps. The former two results provide a rigorous framework for the numerical approximation of random acims using a Fourier-based approach and Ulam's method, respectively; we also demonstrate the efficacy of these schemes.