Stochastic Stability of ACIMs for Piecewise Expanding $C^{1+\varepsilon}$ Maps

TL;DR

This paper proves that absolutely continuous invariant measures for certain piecewise expanding maps are stably preserved under small stochastic perturbations, extending stability results to minimal regularity conditions.

Contribution

It establishes stochastic stability of ACIMs for $C^{1+ ext{epsilon}}$ maps using a new uniform Lasota--Yorke inequality in a generalized BV space.

Findings

Invariant densities $h_ extdelta$ converge to $h$ in $L^1$ as noise diminishes.

Uniform Lasota--Yorke inequality holds in $BV_{1,1/p}$ space for perturbed operators.

Stochastic stability is proven under minimal regularity conditions, where $C^1$ regularity fails.

Abstract

We prove stochastic stability of absolutely continuous invariant measures (ACIMs) for piecewise expanding $C^{1+\varepsilon}$ maps of the interval. For maps $\tau$ in the class $\mathcal{T}([0,1]; s, \varepsilon)$, we consider perturbed Frobenius--Perron operators $P_\delta = Q_\delta P_\tau$, where $Q_\delta$ is a Markov smoothing operator modeling noise of intensity $\delta > 0$. In the generalized bounded variation space $BV_{1,1/p}$, we establish a Lasota--Yorke inequality uniform in $\delta$. Consequently, each $P_\delta$ admits an invariant density $h_\delta \in BV_{1,1/p}$, and $h_\delta \to h$ in $L^1$ as $\delta \to 0$, where $h$ is the ACIM density of $P_\tau$. Our proof combines the $BV_{1,1/p}$ framework, adapted from recent ACIM existence results, with uniform quasi-compactness and perturbation theory for transfer operators. This establishes stochastic stability under minimal $C^{1+\varepsilon}$ regularity ($\varepsilon > 0$), where the $C^1$ case is known to fail.