Contractivity of a Markov operator on the space of normalised positive distributions

The goal of the article is to establish the contractivity, on the space of normalised positive distributions, of a certain class of Markov operators defined by stochastic kernels. The motivation for this work is the promising use of stationary densities in characterising convergence properties of a certain class of discrete-time random algorithms, especially when the so-called associated ordinary differential equation has multiple asymptotically stable equilibria, and no other stable structures.