Approach to Markov operators on spaces of measures by means of equicontinuity

The subject of this thesis, ‘Approach to Markov Operators on Spaces of Measures by Means of Equicontinuity’, combines an analytical and probabilistic approach to Markov operators. We look at Markov operators coming from deterministic dynamical systems and also stochastic processes which come from a probabilistic approach.In the study of Markov operators and Markov semigroups the central problems are to understand the behaviour of the processes and semigroups. Of particular interest is to identify the existence and uniqueness of invariant measures and long term behaviour of the process and dynamical system defined by the associated Markov operator or semigroup. Research on these questions dates back to the works of Andrey Markov, who described a Markov property for chains. A big part of theory for Markov chains can be found in the book by Meyn and Tweedie, who made a big contribution to the theory of Markov chains and gave a noteworthy description of e-chains, which was the motivation to working with equicontinuity properties for many authors. This theory is applicable when the underlying state space is locally compact. If it is not - in the generality of so-called Polish spaces - there is theory under development. Lasota and Szarek, and in recent years Worm generalized theory of Markov operators and families of Markov operators to this setting. In subsequent years, the theory was being developed starting with contractive Markov operators in the works of Lasota, through non-expansive Markov operators in Szarek’s,, and finally equicontinuous families of Markov operators in that of Szarek, Hille and Worm. We extend their results and give a new light to the existing ones by providing less restrictive conditions in cases.