On stationary stochastic flows and Palm probabilities of surface processes

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 $P_{\Phi}$.