Lattices in Classical Potential Theory and Martingale Theory

Abstract

Submartingales martingales and supermartingales are analogs in the context of martingale theory of subharmonic harmonic and superharmonic functions in the context of classical potential theory. The correspondence between these two contexts has two aspects. In the first place many of the manipulations of supermartingales correspond exactly to manipulations of superharmonic functions. This has been exhibited in previous chapters by the common choice of nomenclature, for example, D, S, Sm, LM, GM, \( {{\tau }_{ \bullet }},R_{ \bullet }^{ \bullet } \). In the second place under appropriate hypotheses the composition of a superharmonic function with Brownian motion is a supermartingale; for example, see Section 2.IX.7. In this chapter lattice aspects of classical potential theory and martingale theory will be developed simultaneously.