Differential operators and approximation processes generated by Markov operators

In this survey paper we report some recent results concerning some classes of differential operators as well as some sequences of positive approximation processes which can be constructed by means of a given Markov operator, the main aim being to investigate whether these differential operators are generators of positive semigroups and whether the semigroups can be approximated by iterates of the approximation processes themselves. Among other things, this theory discloses several interesting applications by highlighting, in particular, the relationship among positive semigroups, initial-boundary value problems, approximation theory and Markov processes and by offering a unifying approach to the study of diverse differential problems.