Asymptotic properties of quadratic stochastic operators acting on the L1L1 space

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours and these have been introduced and studied recently in the ℓ1ℓ1 space. It turns out that in principle most of the results can be carried over to the L1L1 space. However, due to topological properties of this space one has to restrict in some situations to kernel quadratic stochastic operators. In this article we study the uniform and strong asymptotic stability of quadratic stochastic operators acting on the L1L1 space in terms of convergence of the associated (linear) nonhomogeneous Markov chains.