Linear Expansion of Isotropic Brownian Flows

Abstract

We consider an isotropic Brownian flow on $R^d$ for $d\geq 2$ with a positive Lyapunov exponent, and show that any nontrivial connected set almost surely contains points whose distance from the origin under the flow grows linearly with time. The speed is bounded below by a fixed constant, which may be computed from the covariance tensor of the flow. This complements earlier work, which showed that stochastic flows with bounded local characteristics and zero drift cannot grow at a linear rate faster than linear.