On stationary stochastic flows and Palm probabilities of surface processes

We consider a random surface . in Rd tessellating the space into cells and a random vector field u which is smooth on each cell but may jump on .. Assuming the pair (.,u) stationary we prove a relationship between the stationary probability measure P and the Palm probability measure P. of P with respect to the random surface measure associated with .. 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 ., 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. 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..