# Strongly mixing smooth planar vector field without asymptotic directions

## TL;DR

This paper constructs a smooth, stationary planar vector field with no asymptotic directions, using a Voronoi tessellation based on a compound Poisson process, demonstrating polynomial mixing properties.

## Contribution

It introduces a novel construction of a vector field with specific mixing and asymptotic properties, expanding understanding of random smooth vector fields.

## Key findings

- Vector field is polynomially mixing
- No integral curves have asymptotic directions
- Construction uses Voronoi tessellation based on a compound Poisson process

## Abstract

We use a Voronoi-type tessellation based on a compound Poisson point process to construct a polynomially mixing stationary random smooth planar vector field with bounded nonnegative components such that, with probability one, none of the associated integral curves possess an asymptotic direction.