Isotropic stochastic flows
Various results on homogeneous isotropic stochastic flows in Euclidean space \({\mathbb{R}}^ d\) are gathered here. Being mainly formulated in terms of a decomposition of the flow into a potential (curl-free) and a solenoidal (divergence-free) part they include characterizations of the asymptotic behaviour of two point distances (which turns out to be different for \(d=2\), \(d=3\) or \(d\geq 4\), respectively) and of angle-length relations between pairs of tangent vectors - e.g., tangent vectors line up asymptotically, which is proved without explicitly using the fact that the two leading Lyapunov exponents are distinct. Further an analogue of the curl-free property is derived for potential isotropic flows and estimates of the lengths of arcs under the flow are given in dependence of their shapes. The paper concludes with a brief discussion of homogeneous nonisotropic flows.
scientific article; zbMATH DE number 3980117