Some Asymptotic Properties of Stochastic Flows

The present work treats several asymptotic properties of stochastic flows in Euclidean space, whose distributions are frequently assumed to be invariant under rotations of the state space. Despite the fact that the study of some of these flows goes back to the 1950's new models are introduced. The first main chapter treats the asymptotic behavior of the shape of the set of points in the plane that has been visited up to some time by a stochastic flow. It is shown in the case of a planar isotropic Brownian flow that this shape is deterministic in probability. The second main result is the extension of the so-called Margulis-Ruelle Inequality to the case of an isotropic Ornstein-Uhlenbeck flow, which is a start to make Pesin theory suitable for the case of a non-compact state space. The last two main chapters are devoted to the asymptotic expansion of the spatial derivative of a stochastic flow taking the supremum over a compact set of initial points in space. It is shown that this expansion is at most exponentially fast in time and a deterministic bound on the expansion speed is obtained.