On stochastic stability of expanding circle maps with neutral fixed points

TL;DR

This paper investigates the stochastic stability of expanding circle maps with neutral fixed points, demonstrating stability for a broader range of parameters than previously known, and showing convergence of stationary measures.

Contribution

It extends the understanding of stochastic stability to cases with lpha in (0,1), where prior results only covered lpha  1, revealing new stability properties.

Findings

Stochastic stability holds for lpha in (0,1).

Stationary measures converge to the deterministic invariant measure as noise diminishes.

The results unify stability behavior across different lpha regimes.

Abstract

It is well-known that the Manneville-Pomeau map with a parabolic fixed point of the form $x\mapsto x+x^{1+\alpha} \mod 1$ is stochastically stable for $\alpha\ge 1$ and the limiting measure is the Dirac measure at the fixed point. In this paper we show that if $\alpha\in (0,1)$ then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero.