On partial diffusion and mixing without hypoellipticity

TL;DR

This work addresses exponential mixing for a degenerate diffusion-transport model $\partial_t u - \partial_{xx}u + V(x)\partial_yu =0$ on $\mathbb{T}^2$, without assuming global hypoellipticity. It develops two complementary routes to quantitative relaxation: (i) a Doeblin-type argument with explicit lower bounds when $V$ has two plateaux (or approximate regularity via a weak local Hörmander condition), and (ii) spectral resolvent estimates in 1D combined with Fourier decomposition in $y$ to obtain $L^2$- and $L^p$-mixing rates that depend on variational quantities $\omega_2(V)$ and $\Osc(V)$. The paper also discusses smoothing limitations in plateau regimes, contrasts with hypocoercivity strategies, and outlines an optimal-control perspective for constructing fundamental solutions, highlighting both its potential and current limitations. Together, these results provide explicit mixing criteria for a broad class of irregular transport fields and clarify how partial diffusion can induce global mixing even in non-hypoelliptic settings.

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.

Theorems & Definitions (10)

• Lemma 1
• Theorem 1
• Theorem 2
• Proposition 1
• Proposition 2
• proof
• Remark 1
• proof : Proof of Proposition \ref{['prop:L2']}
• Proposition 3: Extended from Wei21
• proof : Proof of Proposition \ref{['prop:v2']}