Consistent families of Brownian motions and stochastic flows of kernels

Abstract

Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice ${\mathbf{Z}}$. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on $[0,1]$ and dividing the mass in that proportion. One part then moves to each of the two adjacent sites. This paper considers a continuous analogue of this evolution, which may be described by means of a stochastic flow of kernels, the theory of which was developed by Le Jan and Raimond. One of their results is that such a flow is characterized by specifying its $N$ point motions, which form a consistent family of Brownian motions. This means for each dimension $N$ we have a diffusion in ${\mathbf{R}}^N$, whose $N$ coordinates are all Brownian motions. Any $M$ coordinates taken from the $N$-dimensional process are distributed as the $M$-dimensional process in the family. Moreover, in this setting, the only interactions between coordinates are local: when coordinates differ in value they evolve independently of each other. In this paper we explain how such multidimensional diffusions may be constructed and characterized via martingale problems.