A Deterministic Displacement Theorem for Poisson Processes

We announce a deterministic analog of Bartlett’s displacement theorem. The result is that a Poisson property is stable with respect to deterministic Hamiltonian displacements. While the random point configurations move according to an n-body evolution, the mean measure P satisfies a nonlinear Vlasov type equation Ṗ + y · ∇xP − ∇y · E(P ) = 0. Combined with Bartlett’s theorem, the result generalizes to interacting Brownian particles, where the mean measure satisfies a McKean-Vlasov type diffusion equation Ṗ + y · ∇xP −∇y ·E(P )− c∆P = 0. 1. Poisson processes Some families of distributions in probability theory are natural because they appear in central limit theorems, because they maximize entropy and because they are invariant under convolutions. An example is the Gaussian distribution on R: adding and then normalizing independent random variables of the same distribution give a new distribution with larger relative entropy except at the attractive Gaussian fixed point of this renormalization map. A similar fixed point on discrete distributions is the Poisson distribution on N. While the Gaussian distribution gives rise to stochastic processes like Brownian motion, the Poisson distribution occurs in random point processes like the Poisson process which is defined as follows: let S be some Euclidean space and let P be a finite measure on S. A Poisson process consists of a collection of random finite sets Π(ω) for which the random variables NB(ω) = |Π(ω)∩B| counting the relative number of points in a Borel set B ⊂ S are Poisson distributed with mean P [B] and such that NBj are independent for disjoint sets Bj ⊂ S. The mean measure P determines the process because a typical point set Π(ω) is obtained by picking randomly a natural number d = d(ω) with probability e−P [S](d!)−1, choosing then independently d random points a1(ω), . . . , ad(ω) from S with law P and forming the set Π(ω) = {a1(ω), . . . , ad(ω)}. We call P (ω) the counting measure on the finite set Π(ω). It satisfies P (ω)[B] = NB(ω). Poisson processes occur frequently in applications, for example as models for traffic on freeways, stars in galaxies or populations of plants. What happens if the point sets are evolved in time possibly with interaction? To have the displaced process also as a Poisson process is useful: the mean measure P t at a later time Received by the editors July 28, 1997. 1991 Mathematics Subject Classification. Primary 58F05, 82C22, 60G55; Secondary 70H05, 60K35, 60J60.