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

Abstract

We prove stochastic stability of absolutely continuous invariant measures (ACIMs) for piecewise expanding $C^{1+\varepsilon}$ maps of the interval. For maps $τ$ in the class $\mathcal{T}([0,1]; s, \varepsilon)$, we consider perturbed Frobenius--Perron operators $P_δ= Q_δP_τ$, where $Q_δ$ is a Markov smoothing operator modeling noise of intensity $δ> 0$. In the generalized bounded variation space $BV_{1,1/p}$, we establish a Lasota--Yorke inequality uniform in $δ$. Consequently, each $P_δ$ admits an invariant density $h_δ\in BV_{1,1/p}$, and $h_δ\to h$ in $L^1$ as $δ\to 0$, where $h$ is the ACIM density of $P_τ$. 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.