Exponential Growth Rate for Derivatives of Stochastic Flows

Abstract

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].