The Annals of Probability

The Sharp Markov Property of Levy Sheets

Robert C. Dalang and John B. Walsh

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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.

Article information

Ann. Probab., Volume 20, Number 2 (1992), 591-626.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G60: Random fields
Secondary: 60G55: Point processes 60J75: Jump processes 60J30 60E07: Infinitely divisible distributions; stable distributions 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60H15: Stochastic partial differential equations [See also 35R60]

Levy process Levy sheet Brownian sheet Poisson sheet Markov property sharp field


Dalang, Robert C.; Walsh, John B. The Sharp Markov Property of Levy Sheets. Ann. Probab. 20 (1992), no. 2, 591--626. doi:10.1214/aop/1176989793.

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