# American Institute of Mathematical Sciences

September  2017, 10(3): 799-822. doi: 10.3934/krm.2017032

## On a linear runs and tumbles equation

 1 Paris-Dauphine, Institut Universitaire de France (IUF), PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassignys, 75775 Paris Cedex 16, France 2 Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

* Corresponding author: Stéphane Mischler

Received  February 2016 Revised  September 2016 Published  December 2016

We consider a linear runs and tumbles equation in dimension $d ≥ 1$ 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. [8] in dimension $d=1$. Our analysis is based on the Krein-Rutman theory revisited in [23] 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.

Citation: Stéphane Mischler, Qilong Weng. On a linear runs and tumbles equation. Kinetic & Related Models, 2017, 10 (3) : 799-822. doi: 10.3934/krm.2017032
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##### References:
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