Invariant measures for random piecewise convex maps

We establish the existence of Lebesgue-equivalent conservative and ergodic sigma -finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighbourhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong recurrence to a fixed point.