An important part of the program will consist of a study of Markov semi-groups and propagators, by employing techniques taken from functional analysis and from the theory of stochastic processes. For locally compact spaces there is a one-to-one correspondence between strongly continuous Feller-Dynkin semi-groups and strong Markov processes (with certain continuity properties. Feller-Dynkin semigroups leave the space of bounded continuous functions vanishing at infinity invariant. This correspondence gives rise to an interaction between stochastic analysis and classical operator semi-group theory. However, many interesting spaces are not locally compact. Nevertheless these more general spaces are interesting and important from the viewpoint of stochastic analysis and possibilities for applications. Examples of such spaces are: Wiener space, loop spaces, Fock space. Many of these spaces are Polish. The idea is to develop an analysis which includes these Polish spaces. In this context we also want to investigate the martingale problem. In the commutative case this leads to problems as described in the paper: J.A. van Casteren, Some problems in stochastic analysis and semigroup theory (in Proceedings of the First International Conference of Semigroups of Operators: Theory and Applications, December 1998, Newport Beach, California The main organizer/chairman is A.V. Balakrishnan). Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 42; Publisher: Birkhauser Verlag, Basel, Switzerland, 2000; pp. 43--60. These problems were for a part repeated and further elaborated in J.A. Van Casteren, Markov processes and Feller semigroups "Conferenze del Seminario di Matematica dell'Universita di Bari" Estratti Dalle Conferenze del Seminario di Matematica, Proceedings of the Summer School Operator methods for Evoluton Equations and Approximation Problems (OMEEAP) 2002, Roma 2003, pages 19-93. In the non-commutative situation some other but related problems were described in the same publication (here ``positive'' is replaced with ``completely positive'', and Feller generator is substituted by Lindblad generator). Shortly the book Jan A. Van Casteren, Markov processes, Feller semigroups and evolution equations, 650 pages will be resubmitted for publication. This book contains many of the results described above.