On Stationary Stochastic Ows and Palm Probabilities of Surface Processes

We consider a random surface in R d tesselating the space into cells and a random vector eld u which is smooth on each cell but may jump on. Assuming the pair ((; u) stationary we prove an inversion formula expressing the probability of an event under the stationary probability in terms of the Palm probability P deened by the random surface measure associated with. This result involves the ow of u induced on the individual cells and generalizes a fundamental inversion formula for stationary point processes on the line. We apply this result to derive some general properties of P as, for instance, an invariance property. Under the assumption that the ow can be deened for all time points, we derive conditions that are necessary and suucient for the ow to be volume-preserving. Our nal application deals with the spherical contact distribution in stochastic geometry.