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

Speaker
Luca Ziviani, Universite Paris Dauphine  

Abstract
In this talk we are going to present some recent results about the Run and Tumble equations, a kinetic model for the movement of bacteria subjected to the presence of a chemotactic substance. In the majority of previous works, the set of admissible velocities is bounded and in these cases an exponential trend of convergence towards an equilibrium has been proved. In our work we consider the Run and Tumble equation when the set velocities is the whole space R d and the distribution M of post-reorientation velocities is sub-exponentially or super-exponentially decaying. We are able to prove existence and uniqueness of a steady state and a sub-exponential trend to the equilibrium. Moreover, thanks to semigroup theory, we are able to prove some L∞ estimate on the steady state, which highlight how the decay of the steady state in the spatial variable x depends on its decay in the velocity variable v.