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

Abstract

We prove stochastic stability of absolutely continuous invariant measures (ACIMs) for piecewise expanding $C^{1+\epsilon }$ maps of the interval. For maps $\tau$ in the class $\mathcal{T}([0,1];s,\epsilon )$, 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 $B{V}_{1,1/p}$, we establish a Lasota--Yorke inequality uniform in $\delta$. Consequently, each $P_{\delta }$ admits an invariant density $h_{\delta }\in B{V}_{1,1/p}$, 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 $B{V}_{1,1/p}$ framework, adapted from recent ACIM existence results, with uniform quasi-compactness and perturbation theory for transfer operators. This establishes stochastic stability under minimal $C^{1+\epsilon }$ regularity ($\epsilon >0$), where the $C^1$ case is known to fail.