Funcionais aditivos de processos de exclusão

Neste trabalho consideram-se processos de exclusão partindo da medida de Bernoulli produto. Obtém-se o limite em escala de funcionais aditivos, como por exemplo do tempo de ocupação, a partir das flutuações da densidade.In this work we consider exclusion processes starting from the Bernoulli product measure. We obtain the scaling limits of additive functionals, as for example the occupation time, from the density fluctuations.Fundação para a Ciência e a Tecnologia (FCT

Equilibrium fluctuations for totally asymmetric particle systems

Fundação Calouste GulbenkianFundação para a Ciência e a Tecnologia (FCT

Microscopic dynamics for the porous medium equation

In this work, I present an interacting particle system whose dynamics conserves the total number of particles but with gradient transition rates that vanish for some configurations. As a consequence, the invariant pieces of the system, namely, the hyperplanes with a fixed number of particles can be decomposed into an irreducible set of configurations plus isolated configurations that do not evolve under the dynamics. By taking initial profiles smooth enough and bounded away from zero and one and for parabolic time scales, the macroscopic density profile evolves according to the porous medium equation. Perturbing slightly the microscopic dynamics in order to remove the degeneracy of the rates the same result can be obtained for more general initial profiles

Simple exclusion process : from randomness to determinism

In this work I introduce a classical example of an Interacting Particle System: the Simple Exclusion Process. I present the notion of hydrodynamic limit, which is a Law of Large Numbers for the empirical measure and an heuristic argument to derive from the microscopic dynamics between particles a partial differential equation describing the evolution of the density profile. For the Simple Exclusion Process, in the Symmetric case ($p=1/2$) we will get to the heat equation while in the Asymmetric case ($p\neq{1/2}$) to the inviscid Burgers equation. Finally, I introduce the Central Limit Theorem for the empirical measure and the limiting process turns out to be a solution of a stochastic partial differential equation