  Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic
flows which has been studied extensively by various authors. Their rich
structure allows for explicit calculations in several situations and makes them
a natural object to start with if one wants to study more general stochastic
flows. Often the intuition gained by understanding the problem in the context
of IBFs transfers to more general situations. However, the obvious link between
stochastic flows, random dynamical systems and ergodic theory cannot be
exploited in its full strength as the IBF does not have an invariant
probability measure but rather an infinite one. Isotropic Ornstein-Uhlenbeck
flows are in a sense localized IBFs and do have an invariant probability
measure. The imposed linear drift destroys the translation invariance of the
IBF, but many other important structure properties like the Markov property of
the distance process remain valid and allow for explicit calculations in
certain situations. The fact that isotropic Ornstein-Uhlenbeck flows have
invariant probability measures allows one to apply techniques from random
dynamical systems theory. We demonstrate this by applying the results of
Ledrappier and Young to calculate the Hausdorff dimension of the statistical
equilibrium of an isotropic Ornstein-Uhlenbeck flow.
