Markov Operators, Positive Semigroups and Approximation Processes

During the last twenty years important progresses have been made, from the point of view of constructive approximation theory, in the study of initial-boundary value differential problems of parabolic type governed by positive 0-semigroups of operators. The main aim of this approach is to construct suitable positive approximation processes whose iterates strongly converge to the semigroups which, as it is well-known, in principle furnish the solutions to the relevant initial-boundary value differential problems. By means of such kind of approximation it is then possible to investigate, among other things, preservation properties and the asymptotic behavior of the semigroups, i.e., spatial regularity properties and asymptotic behavior of the solutions to the differential problems. This series of researches, for a survey on which we refer to [25] and the references therein, has its roots in several studies developed between 1989 and 1994 which are documented in Chapter 6 of the monograph [18]. These studies were concerned with special classes of second-order elliptic differential operators acting on spaces of smooth functions on finite dimensional compact convex subsets, which are generated by a positive projection. The projections themselves offer the tools to construct an approximation process whose iterates converge to the relevant semigroup, making possible the development of a qualitative analysis as above. This theory disclosed several interesting applications by stressing the relationship among positive semigroups, initial-boundary value problems, Markov processes and approximation theory, and by offering, among other things, a unifying approach to the study of diverse differential problems. Nevertheless, over the subsequent years, it has naturally arisen the need to extend the theory by developing a parallel one for positive operators rather than for positive projections and by trying to include in the same project of investigations more general differential operators having a first-order term. The aim of this research monograph is to accomplish such an attempt by considering complete second-order (degenerate) elliptic differential operators whose leading coefficients are generated by a positive operator, by means of which it is possible to construct suitable approximation processes which approximate the relevant semigroups. Some aspects of the theory are treated also in infinite dimensional settings. viii Preface The above described more general framework guarantees to notably enlarge the class of differential operators by including, in particular, those which can be obtained by usual operations with positive operators such as convex combinations, compositions, tensor products and so on. Moreover, this non-trivial generalization discloses new challenging problems as well. However, the approximation processes which we construct in terms of the given positive operator seem to have an interest in their own also for the approximation of continuous functions. For these reasons, a special emphasis is placed upon various aspects of this theme as well.