The Sharp Markov Property of Levy Sheets

This paper examines the question of when a two-parameter process X of independent increments will have Levy's sharp Markov property relative to a given domain D. This property states intuitively that the values of the process inside D and outside D are conditionally independent given the values of the process on the boundary of D. Under mild assumptions, X is the sum of a continuous Gaussian process and an independent jump process. We show that if X satisfies Levy's sharp Markov property, so do both the Gaussian and the jump process. The Gaussian case has been studied in a previous paper by the same authors. Here, we examine the case where X is a jump process. The presence of discontinuities requires a new formulation of the sharp Markov property. The main result is that a jump process satisfies the sharp Markov property for all bounded open sets. This proves a generalization of a conjecture of Carnal and Walsh concerning the Poisson sheet.