---
tag: YS7RF1PE42
kind: concept
name: Axiomatic potential theory
msc: [31D05]
specializes: [9PM8Q2E30Q]
sources:
  - url: https://en.wikipedia.org/wiki/Potential_theory
---

## Summary

Axiomatic potential theory studies [[potential theory|9PM8Q2E30Q]] 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.

## Explanations

### L1 human

Classical potential theory is built on the Laplace equation, but many of its theorems use only a few of its features.
Axiomatic potential theory picks out those features and takes them as axioms.
Any equation or random process that satisfies the axioms then inherits the whole theory at once.
The main frameworks are the [[harmonic spaces|DWN0XEQHHK]] of Brelot and of Bauer, and later more abstract ones such as [[balayage spaces|G9M1H5NNF6]] and [[H-cones|Q5C1E10CSQ]].
For example, the solutions of the heat equation fit Bauer's axioms, so results like the minimum principle for supersolutions follow from the general theory.