Sub-exponential tails in biased run and tumble equations with unbounded velocities

TL;DR

This paper studies run and tumble equations with unbounded velocities, demonstrating existence, uniqueness, and convergence to a steady state with sub-exponential tails, extending previous bounded velocity results.

Contribution

It introduces new analysis techniques to handle unbounded velocities, showing sub-exponential tail behavior and convergence rates in this setting.

Findings

Existence and uniqueness of solutions with unbounded velocities

Sub-exponential tails in the equilibrium distribution

Sub-exponential convergence rate to steady state

Abstract

Run and tumble equations are widely used models for bacterial chemotaxis. In this paper, we are interested in the long time behaviour of run and tumble equations with unbounded velocities. We show existence, uniqueness and quantitative convergence towards a steady state. In contrast to the bounded velocity case, the equilibrium has sub-exponential tails and we have sub-exponential rate of convergence to equilibrium. This produces additional technical challenges. We are able to successfully adapt both Harris' type and $\sfL^2-$ hypocoercivity \textit{a la} Dolbeault-Mouhot-Schmeiser techniques.