Stochastic perturbations and Ulam's method for W-shaped maps (Q1948781)

The authors study the absolutely continuous invariant measures (ACIMs) of piecewise expanding maps on the interval, and prove their stability under stochastic perturbations. That is, if the map satisfies the harmonic slope condition (\(\frac{1}{s_i}+\frac{1}{s_{i+1}}<1\) for all \(i\)), then the ACIM is stable under all small stochastic perturbations \(Q\) that do not increase the variation: \(\bigvee Qf\leq \bigvee f\). The proof relies on a stronger Lasota-Yorke inequality proved in [\textit{P. Eslami} and \textit{P. Góra}, Proc. Am. Math. Soc. 141, No. 12, 4249--4260 (2013; Zbl 1366.37095)]. See also [\textit{C. Liverani}, Ergodic Theory Dyn. Syst. 33, No. 1, 168--182 (2013; Zbl 1268.37026)]. In contrast to the unstable behavior of the ACIM under deterministic perturbations of the \(W\)-map, the authors also prove the stability of the \(W\)-map under small stochastic perturbations by some direct calculations.