Open Access
May 2000 On stationary stochastic flows and Palm probabilities of surface processes
G. Last, R. Schassberger
Ann. Appl. Probab. 10(2): 463-492 (May 2000). DOI: 10.1214/aoap/1019487351

Abstract

We consider a random surface Φ in Rd tessellating the space into cells and a random vector field u which is smooth on each cell but may jump on Φ. Assuming the pair (Φ,u) stationary we prove a relationship between the stationary probability measure P and the Palm probability measure PΦ of P with respect to the random surface measure associated with Φ. This result involves the flow of u induced on the individual cells and generalizes a well-known inversion formula for stationary point processes on the line. An immediate consequence of this result is a formula for certain generalized contact distribution functions of Φ, and as first application we prove a result on the spherical contact distribution in stochastic geometry. As another application we prove an invariance property for PΦ which again generalizes a corresponding property in dimension d=1. Under the assumption that the flow can be defined for all time points, we consider the point process N of sucessive crossing times starting in the origin 0. If the flow is volume preserving, then N is stationary and we express its Palm probability in terms of PΦ.

Citation

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G. Last. R. Schassberger. "On stationary stochastic flows and Palm probabilities of surface processes." Ann. Appl. Probab. 10 (2) 463 - 492, May 2000. https://doi.org/10.1214/aoap/1019487351

Information

Published: May 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1052.60011
MathSciNet: MR1768238
Digital Object Identifier: 10.1214/aoap/1019487351

Subjects:
Primary: 60D05 , 60G60

Keywords: Palm probability , point process , Random field , random measure , random set , Random surface , stochastic flow , Stochastic geometry

Rights: Copyright © 2000 Institute of Mathematical Statistics

Vol.10 • No. 2 • May 2000
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