Introduction to Potential Theory via Applications

TL;DR

This paper introduces fundamental concepts of potential theory, including subharmonic functions, potentials, and applications, aiming to clarify the basic ideas and their relevance in various mathematical contexts for readers with a background in analysis and probability.

Contribution

It provides a clear, accessible exposition of potential theory concepts and applications, with a slight deviation from traditional presentations to enhance understanding.

Findings

Detailed explanation of the Riesz decomposition theorem

Illustration of the Dirichlet problem and harmonic measure

Application of Green's function to polynomial growth

Abstract

We introduce the basic concepts related to subharmonic functions and potentials, mainly for the case of the complex plane and prove the Riesz decomposition theorem. Beyond the elementary facts of the theory we deviate slightly from the usual path of exposition and introduce further concepts alongside with applications. We cover the Dirichlet problem in detail and illustrate the relations between potential theory and probability by considering harmonic measure and its relation to Brownian motion. Furthermore Green's function is introduced and an application to growth of polynomials is given. Equilibrium measures are motivated by their original development in physics and we end with a brief discussion of capacity and its relation to Hausdorff measure. We hope that the reader, who is familiar with the main elements of real analysis, complex analysis, measure theory and some probability theory benefits from these notes. No new results are presented but we hope that the style of presentation enables the reader to understand quickly the basic ideas of potential theory and how it can be used in different contexts. The notes can also be used for a short course on potential theory. Therefore the required prerequisites are described in the Appendix. References are given where expositions and details can be found; roughly speaking, familiarity with the basic foundations of real and complex analysis should suffice to proceed without any background reading.