Harmonic Spaces and Associated Markov Processes

Abstract

These lectures should be understood as an introduction (mainly for non-specialists) to one example of a so-called axiomatic potential theory, namely the theory of harmonic spaces and to the relations of this theory with the theory of Markov processes. The notion of a harmonic space arose from the study of elliptic and parabolic linear differential equations. Potential-theoretic aspects of the theory of Markov processes have their origin in the study of Brownian motion. This particular Markov processes led to probabilistic interpretations of many facts from classical potential theory. Many of these interpretations will be proved here in the homework of harmonic spaces.

The lectures are organized as follows; After a short introduction to the notion of a harmonic space, we present in a very condensed form parts of the theory of these spaces. We then describe the construction of an associated semigroup (P_t)_t≥0 of kernels and their interpretation as the transition semigroup of a Markov process. Then a collection of important notions and results from the theory of Markov processes follows. In a final paragraph, the most important potential-theoretic notions find a probabilistic interpretation.