Invariant measures for random transformations expanding on average

The thermodynamic formalism for random transformations expanding on average is revisited. We consider the associated random transfer operators with Hölder continuous random potentials, and prove a random version of the Ruelle–Perron–Frobenius Theorem. This result allows us to construct random invariant measures for the transformations considered. These measures are ergodic, and enjoy fiberwise exponential decay of correlations. As a method of proof, we construct a family of cones of positive functions for which the transfer operator is a strict contraction. Application of a random fixed point theorem then yields a maximal random eigenvalue and eigenvector of the transfer operator.