Mots clés : systèmes de particules, processus de réaction-diffusion, limites hydrodynamiques

Abstract

The subject of this thesis is related to infinite particle systems. First we study a reaction-diffusion process with two species, called hawks and doves. It is a modelisation of the evolution of two animal species living in interaction on the same domain. On the lattice Z^d, some hawks and doves evolve in time : they move on the lattice and births and deaths occur at random rates. The first results concern the control of the moments and the invariant measures. Afterwards, we obtain the hydrodynamic limits of the process (passage between the microscopic discrete scale and the continuous one), to deduce partial differential equations for the evolution of population densities. Two techniques are used : One is based on the correlation functions of the process, the second is the relative entropy method (first in finite volume, then extended to infinite volume by coupling).

In the second part, we study the non gradient generalized exclusion process, via the H_-1 norm and we obtain a strong version of the hydrodynamic limits in infinite volume.

Keywords: reaction-diffusion processes, hydrodynamic limits, interacting particle systems