The Probabilistic Representation of Positive Solutions

Abstract

In this chapter, we address the general problem of providing a probabilistic classification of positive solutions to the partial differential equation △u = u²in a smooth domain. We give a complete solution to this problem in the case of the planar unit disk. Precisely, we show that solutions are in oneto-correspondence with their traces, where the trace of a solution consists of a compact subset of the boundary and a Radon measure on the complement of this compact subset in the boundary. Furthermore, we give an explicit probabilistic formula for the solution associated with a given trace. At the end of the chapter, we discuss extensions to higher dimensions or more general equations.