SUMMARYWe start with a stochastic flow of diffeomorphisms of the sphce. Particles enter the space at random times and places. Each particle is carried by the flow for some random amount of time. We examine the point process formed by the particles at a fixed time, on the evolution of that point process as time varies, and on the equilibrium law of the point process.KEY WORDS Stochastic flows Mass transport Statistical equilibrium Birth and death processes
. INTRODUCTIONRecent years have brought renewed interest and many advances in stochastic flows and their ergodic theory. Most studies have been concentrated on characterizations of important classes of flows, the effect of a flow on a spatially distributed mass, and the ergodic behaviour of that mass distribution. Our aim here is to extend such studies to the cases where there is creation and annihilation of mass over time (in addition to the motion on the flow).For visualization purposes, we think of a turbulent flow of some fluid on R2. Certain pollutant particles are present, at time 0, at various random locations. Additional particles enter at various random times and places. Each particle is carried by the flow for some random time and eventually disappears owing to sedimentation or chemical decay. We are interested in the point process on R z formed by the positions of particles that are present at time f, the evolution of that point process as t varies, and the limiting probability law of that point process as t tends to infinity. The novel feature here seems to be the immense dependence between the particle paths, which comes from the sameness of the random noise that drives the differential equations defining the turbulent. flow.We describe this particle system in Section 2, give some results on its dynamics in Section 3, and give a very special ergodic limit theorem in Section 4 for its probability law. For the remainder of this section, we put down some preliminaries on stochastic flows and illustrate the effect of a flow on a mass distribution by two examples: one example is for a diffuse mass without creation or annihilation, and the other is for a particle system with annihilation.
Stochastic flows
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