Some Problems in Stochastic Analysis and Semigroup Theory

Abstract

For locally compact spaces there is a one-to-one correspondence between Feller semigroups (those are semigroups which send continuous functions to continuous ones) and (strong) Markov processes. This leads to an interaction between stochastic analysis and classical semigroup theory. However, many interesting topological spaces are not locally compact. Nevertheless from the point of view of (stochastic) analysis and possible applications these more general topological spaces are important. Examples of such spaces are Wiener space, Loop space, Fock space. These spaces are Polish spaces or more general Lusin spaces. We like to develop an analysis which encompasses this kind of spaces. We also like to bring in the martingale problem. In the commutative setting this leads to the problems sketched in §1. In the non-commutative setting some other problems are described in §5 as well. In §1 and in §2 a relationship is established between reciprocal Markov processes and Euclidean quantum mechanics. Moreover there should be a connection with Feynman propagators. The problem proposed in §3 is closely related to work done by Demuth and van Casteren [8]. In order to discuss Neumann scattering the problems proposed in §4 are included. In particular cases these can certainly be solved. In our opinion the described problems are feasible.