Isotropic Ornstein–Uhlenbeck flows

Abstract

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 (IOUFs) 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 IOUFs 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 IOUF.

Characterizing Gaussian Flows Arising from Itô’s Stochastic Differential Equations

“…In certain situations such differences were shown to be diffusions (see [22,Proposition 2.2]). (iv) Semimartingales with independent increments have been considered in [11, Chapter II].…”

Ruelle's Inequality for Isotropic Ornstein–uhlenbeck Flows

“…This class of stochastic flows has been defined in [7] and has been further studied in [5]. Care has to be taken concerning the name "isotropic" because IOUFs (or their laws respectively) are not isotropic at all: the law of an IOUF changes if one applies a translation on the state space and the origin is distinguished by being the only zero of the drift in the SDE.…”

“…We stick to the name because the most important ingredient of an IOUF is an isotropic covariance tensor (see [6]). An IOUF is the flow-version of an Ornstein-Uhlenbeck-process (see [5] for more details) and can be obtained from an Isotropic Brownian Flow (IBF) (see [6] or [13]) via adding a quadratic potential. Definition 2.1.…”

“…IOUFs have been studied in [7] and in [5] and we will recall some facts that we will use later. (2) The one-point motion is an Ornstein-Uhlenbeck-diffusion with generator…”

Asymptotic Growth of Spatial Derivatives of Isotropic Flows

“…In the following we will consider the case where the flow is replaced by its spatial derivative. We emphasize that we consider the asymptotics in time (the spatial asymptotics for a fixed time horizon have been considered in [8] in a very general setting and in [2] in the particular one treated here).…”