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The Sharp Markov Property of Levy Sheets
- Source :
- Ann. Probab. 20, no. 2 (1992), 591-626
- Publication Year :
- 1992
- Publisher :
- Institute of Mathematical Statistics, 1992.
-
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.
- Subjects :
- Statistics and Probability
Field (physics)
Markov property
splitting field
Poisson distribution
Lévy process
Poisson sheet
symbols.namesake
60E07
35R60
Calculus
Statistical physics
Levy process
Brownian motion
Mathematics
60G60
Levy sheet
Splitting field
Brownian sheet
field
sharp
two-parameter Lévy processes
60H15
symbols
60G55
60J30
sharp field
60J75
Statistics, Probability and Uncertainty
Subjects
Details
- ISSN :
- 00911798
- Volume :
- 20
- Database :
- OpenAIRE
- Journal :
- The Annals of Probability
- Accession number :
- edsair.doi.dedup.....18d14883772bf3a2dcf26081961b7c57