Special Examples of Diffusions in Random Environment

TL;DR

This paper presents surprising examples of diffusions in random environments, showing how small changes can lead to loss or reversal of ballistic behavior despite non-zero local drift.

Contribution

It introduces new examples of diffusions with unexpected asymptotic behaviors, highlighting the complex effects of environment perturbations on diffusion dynamics.

Findings

Diffusions with non-zero local drift can lose ballistic behavior.

Small perturbations can reverse the limiting velocity direction.

Examples demonstrate complex asymptotic behaviors in random environments.

Abstract

In this note we present some examples of diffusions in random environment whose asymptotic behavior is rather surprising. We construct a family of diffusions that are small perturbations of Brownian motion with non-vanishing expected local drift under the static measure of the environment but where the ballistic behavior is lost. As slight modifications of this collection of diffusions we also provide examples with ballistic behavior where the non-vanishing limiting velocity points to a direction opposite to the expected local drift under the static measure.