Stochastic stability for weakly hyperbolic contracting Lorenz maps

TL;DR

This paper proves stochastic stability for certain contracting Lorenz maps under random perturbations, showing that stationary measure densities converge to the unperturbed map's physical measure in L^1 norm.

Contribution

It establishes strong stochastic stability results for weakly hyperbolic Lorenz maps, extending previous work and providing convergence of stationary densities.

Findings

Stationary measure densities converge in L^1 norm to the physical measure.

Results apply to maps satisfying the summability condition of exponent 1.

General conditions on maps and perturbations ensure stochastic stability.

Abstract

In this article we study the expanding properties of random perturbations of contracting Lorenz maps satisfying the summability condition of exponent 1. Under general conditions on the maps and perturbation types, we prove stochastic stability in the strong sense: convergence of the densities of the stationary measures to the density of the physical measure of the unperturbed map in the $L^1$-norm. This improves the main result in \cite{Me}.