Exponential growth rate for derivatives of stochastic flows

TL;DR

This paper establishes that for many stochastic flows, the spatial derivatives grow at most exponentially, providing explicit bounds based on local flow characteristics and the set's box dimension.

Contribution

It introduces explicit bounds on the exponential growth rates of derivatives in stochastic flows, linking them to local properties and set dimensions.

Findings

Derivatives grow at most exponentially in stochastic flows.

Explicit bounds depend on local flow characteristics.

Growth rates relate to the box dimension of initial sets.

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.