Differential operators and approximation processes generated by Markov operators

In recent years several investigations have been devoted to the study of large classes of (mainly degenerate) initial-boundary value evolution problems in connection with the possibility to obtain a constructive approximation of the associated positive C0- semigroups by means of iterates of suitable positive linear operators which also constitute approximation processes in the underlying Banach function space. Usually, as a consequence of a careful analysis of the preservation properties of the approximating operators, such as monotonicity, convexity, Hölder continuity and so on, it is possible to infer similar preservation properties for the relevant semigroups and, in turn, some spatial regularity properties of the solutions to the evolution problems. More recently, by continuing along these directions, we started a research project in order to investigate the possibility of associating to a given Markov operator on the Banach space C(K) of all real functions defined on a convex compact subset K of Rd (d \geq1) some classes of differential operators as well as some suitable positive approximation processes. The main aim is 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. By means of a qualitative study of the approximation processes, the approximation formulas could guarantee a similar qualitative analysis of the positive semigroups and, consequently, of the solutions to the evolution equations governed by them. In this survey paper we report some of the main ideas and results we have developed in this respect and which are documented in many papers. The differential operators considered within the framework of the theory fall into classes of operators of wide interest in the theory of evolution equations and in models of population dynamics and mathematical finance. The generation problems for these differential operators have been also studied with other methods which, however, do not allow to get spatial regularity properties of the solutions as well as information about their asymptotic behavior whereas these aspects are successfully obtained with the methods of approximation by positive operators. Furthermore, the involved approximation processes are inspired by some classical ones and, among other things, they generalize the Bernstein operators and the Kantorovich operators in all one-dimensional and multi-dimensional convex domains on which the latter have been considered. These approximation processes seem to have an interest in their own also for the approximation of continuous functions and, in some cases, of p-th power integrable functions. For these reasons their study has been also deepened from several points of view of the approximation theory. The paper contains some noteworthy examples which offer a short outline of the possible application of the theory and show that diverse differential problems scattered in the literature can be encompassed in the present unifying approach.