A Green Proof of Fatou's Theorem

Abstract

This paper is an exposition of some applications of Stochastic Processes to boundary behavior problems for harmonic functions. As an illustration, we give a proof of Fatou's theorem in simply connected plane domains which is probabilistic and does not use the Riemann mapping theorem. The paper closes with some remarks on further related work and open questions.

Key Words:

• Stochastic methods in potential theory
• Fatou's theorem
• Boundary behaviour of harmonic functions