Résumé en anglais

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.