The Sharp Markov Property of Levy Sheets

Abstract

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.