Topics in Energy and Potential Theory

Abstract

Energy is a frustratingly delicate item in modern Markov process theory. It is a subject which commands attention, since it is linked so closely with maximum principles and Hunt’s hypothesis (H). It entered potential theory in the work of Cartan and Dény, where it enabled them to prove various delicate principles about symmetric potential kernels. It has flourished in the modern theory of Dirichlet spaces and has added to the body of knowledge concerning symmetric Markov processes. But while energy is a natural and cooperative partner in the study of symmetric potential kernels, it becomes increasingly intractable as one attempts to study more asymmetric kernels and processes. Concrete results in this domain are few. We present some topics in energy and potential theory for Markov processes with nonsymmetric potential kernels which complement several results and articles by various authors.

Research supported in part by NSF Grant MCS-8002659.