On partial diffusion and mixing without hypoellipticity

TL;DR

This paper studies a Markov process combining diffusion and transport, providing mixing rate estimates under minimal assumptions, even when the transport field is highly irregular or degenerate, without relying on hypoellipticity.

Contribution

It introduces a method to estimate mixing rates for a Markov process with limited regularity assumptions on the transport component, extending beyond hypoelliptic frameworks.

Findings

Quantitative mixing rates are established under weak assumptions.

The approach handles highly irregular and degenerate transport fields.

Results do not require hypoellipticity conditions.

Abstract

A simple Markov process is considered involving a diffusion in one direction and a transport in a transverse direction. Quantitative mixing rate estimates are obtained with limited assumptions about the transport field, which might be highly irregular and/or highly degenerate, in particular quite far from satisfying an hypoellipticity type assumption.