Quenched stochastic stability for eventually expanding-on-average random interval map cocycles

The paper [FGTQ14] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations, including “Ulam’s method”, a popular numerical method for approximating acims. The expansivity requirements of [FGTQ14] were that the cocycle (or powers of the cocycle) should be “expanding on average” before applying a perturbation, such as Ulam’s method. In the present work we make a significant theoretical and computational weakening of the expansivity hypotheses of [FGTQ14], requiring only that the cocycle be eventually expanding on average, and importantly, allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota-Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or “physical” random acims.