Stochastic flows and measure-valued processes

Abstract

This article presents results about one-dimensional Brownian stochastic flows obtained in the department of the random processes during the last 15 years. One-dimensional stochastic flows describe the mutual mo- tion of diffusion particles on the real line. As a good example of such flows, the flows of solutions to SDE can serve. But there exists a large set of flows that can not be obtained from SDE because of the possibility for particles to coalesce. The survey is concentrated mainly on such flows. Here we dis- cuss questions of two types. The first group is devoted to traditional SDE problems like large deviations, discrete approximations, Krylov–Veretennikov expansion, Girsanov theorem. We present corresponding results which have new features due to coalescence phenomena. Another question is related to point structure, which arises in a coalescing set of Brownian motions. Here are the properties of corresponding point measures are discussed.