On stationary stochastic flows and palm probabilities of surface processes

We consider a random surface Phi in R-d tessellating the space into cells a nd a random vector held 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 res ult 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 generalize d contact distribution functions of Phi, and as first application we prove a result on the spherical contact distribution in stochastic geometry. As a nother application we prove an invariance property for P-Phi which again ge neralizes a corresponding property in dimension d = 1. Under the assumption that the flow can be defined for all time points, we consider the point pr ocess N of sucessive crossing times starting in the origin 0. If the flow i s volume preserving then N is stationary and we express its Palm probabilit y in terms of P-Phi.