On a linear runs and tumbles equation

TL;DR

This paper proves the existence, uniqueness, and stability of a steady state for a linear runs and tumbles equation in multiple dimensions, extending previous one-dimensional results using advanced spectral and regularity techniques.

Contribution

It extends the analysis of the runs and tumbles equation to higher dimensions and introduces new moment estimates and regularity methods for the study of its steady states.

Findings

Existence of a unique positive steady state in dimension d ≥ 1.

Asymptotic stability of the steady state.

Enhanced analytical techniques for higher-dimensional runs and tumbles equations.

Abstract

We consider a linear runs and tumbles equation in dimension d $\ge$ 1 for which we establish the existence of a unique positive and normalized steady state as well as its asymptotic stability, improving similar results obtained by Calvez et al. [5] in dimension d = 1. Our analysis is based on the Krein-Rutman theory revisited in [18] together with some new moment estimates for proving confinement mechanism as well as dispersion, multiplicator and averaging lemma arguments for proving some regularity property of suitable iterated averaging quantities.