Weakly mixing smooth planar vector field without asymptotic directions

TL;DR

This paper constructs a smooth, weakly mixing planar vector field with no asymptotic directions, demonstrating complex flow behavior and challenging assumptions about particle trajectories in such fields.

Contribution

It introduces a novel example of a weakly mixing vector field with no asymptotic directions, advancing understanding of flow dynamics in random vector fields.

Findings

Flow has no asymptotic direction with probability 1

Trajectories' limiting directions span the positive quadrant

Weakly mixing fields can lack asymptotic average velocity

Abstract

We construct a planar smooth weakly mixing stationary random vector field with nonnegative components such that, with probability 1, the flow generated by this vector field does not have an asymptotic direction. Moreover, for all individual trajectories, the set of partial limiting directions coincides with those spanning the positive quadrant. A modified example shows that a particle in space-time weakly mixing positive velocity field does not necessarily have an asymptotic average velocity.