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April, 1992 The Sharp Markov Property of Levy Sheets
Robert C. Dalang, John B. Walsh
Ann. Probab. 20(2): 591-626 (April, 1992). DOI: 10.1214/aop/1176989793


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.


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Robert C. Dalang. John B. Walsh. "The Sharp Markov Property of Levy Sheets." Ann. Probab. 20 (2) 591 - 626, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0783.60049
MathSciNet: MR1159561
Digital Object Identifier: 10.1214/aop/1176989793

Primary: 60G60
Secondary: 35R60 , 60E07 , 60G55 , 60H15 , 60J30 , 60J75

Keywords: Brownian sheet , Levy process , Levy sheet , Markov property , Poisson sheet , sharp field

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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