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 $\tau$ 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.