Axiomatic potential theory

Summary

Axiomatic potential theory studies potential theory from a short list of axioms on a sheaf of "harmonic" functions on a locally compact space. The axioms keep the solvability of the Dirichlet problem on small sets and a convergence property of increasing sequences. From them one recovers superharmonic functions, potentials, balayage, the fine topology and the Dirichlet problem. The theory covers the Laplace equation, more general elliptic equations, the heat equation and Markov processes in one framework.