It has been shown by various authors under different assumptions that the diameter of a bounded non-trivial set γ under the action of a stochastic flow grows linearly in time. We show that the asymptotic linear expansion speed if properly defined is deterministic i.e. we show for a 2-dimensional isotropic Brownian flow Φ with a positive Lyapunov exponent that there exists a non-random set B such that we have for > 0, arbitrary connected γ ⊂⊂ R 2 consisting of at least two different points and arbitrarily large times T thatDefinition 1.2 A function b : R d → R d×d is an isotropic covariance tensor if x → b(x) is C ∞ and all derivatives of any order are bounded, b(0) = E d (the d-dimensional unit matrix), x → b(x) is not constant and b(x) = O * b(Ox)O holds for any x ∈ R d and any orthogonal matrix O.
Ruelle's inequality asserts that the entropy of a dynamical system is bounded from above by the Lyapunov characteristic numbers counted with their multiplicities. We show that this inequality holds true in the case of a random dynamical system deduced from an isotropic Ornstein-Uhlenbeck-flow (IOUF).
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.
We show that for a large class of stochastic flows the spatial derivative grows at most exponentially fast even if one takes the supremum over a bounded set of initial points. We derive explicit bounds on the growth rates that depend on the local characteristics of the flow and the box dimension of the set.Remark 3.3. The formulas in Theorem 3.2 still contain the numbers α 2 , α 3 , β 1 and β 3 . Since α 1 , β 2 and β 4 do not appear in the formulas, it is possible to choose α 2 = α 3 = β 1 = β 3 = 2 but a different choice may result in a sharper bound.Remark 3.4. One can establish a corresponding result also in case d = 1 by adjustinĝ c just like we adjustedk in order to extend the moment bound on the differences of derivatives from p ≥ 2 to p > d = 1.Remark 3.5. It is not hard to establish a version of both Lemma 3.1 and Theorem 3.2 for higher derivatives of a stochastic flow using an induction proof along the lines of [6, Proposition 2.3].
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