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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 $\Phi$ in $\mathbb{R}^d$ tessellating the space into cells and a random vector field $u$ which is smooth on each cell but may jump on $\Phi$. Assuming the pair $(\Phi, u)$ stationary we prove a relationship between the stationary probability measure $P$ and the Palm probability measure $P_{\Phi}$ of $P$ with respect to the random surface measure associated with $\Phi$. 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 $\Phi$, 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 $P_{\Phi}$ 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 $P_{\Phi}$.

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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

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Vol.10 • No. 2 • May 2000
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