Brownian Motions on the Homeomorphisms of the Plane

Abstract

Let $z$ denote a point of $R_2$. We study random flows $Z_{st}(z) \in R_2, 0 \leq s \leq t < \infty, z \in R_2, Z_{tu}(Z_{st}(z)) = Z_{su}(z)$ for $s \leq t \leq u$. Such flows are called Brownian if $Z$ is continuous in $(s, t, z)$ and has appropriate spatial and temporal homogeneity properties and if $Z_{st}, Z_{uv}, \cdots$ are independent homeomorphisms of $R_2$ onto $R_2$ when $s \leq t \leq u \leq v \leq \cdots$. For a Brownian flow the coordinates of any $k$ points are a $2k$-dimensional continuous Markov process, $k = 1, 2, \cdots$. If these processes are diffusions whose diffusion matrices have bounded continuous derivatives of order $\leq 2$ (i.e., are $C^2$-bounded), then the diffusion matrices are necessarily obtained in a certain way from the covariance tensor of the field of infinitesimal displacements. A converse is given in the incompressible isotropic case: given a $C^2$-bounded covariance tensor of an isotropic solenoidal $R_2$-valued field in $R_2$, there exists a corresponding incompressible isotropic Brownian flow.