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April, 1981 Brownian Motions on the Homeomorphisms of the Plane
Theodore E. Harris
Ann. Probab. 9(2): 232-254 (April, 1981). DOI: 10.1214/aop/1176994465


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.


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Theodore E. Harris. "Brownian Motions on the Homeomorphisms of the Plane." Ann. Probab. 9 (2) 232 - 254, April, 1981.


Published: April, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0457.60013
MathSciNet: MR606986
Digital Object Identifier: 10.1214/aop/1176994465

Primary: 60B99
Secondary: 60G99

Keywords: diffusion , Flows , Random fields , random homeomorphisms

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 2 • April, 1981
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