Milton Jara
Scaling limits of additive functionals of interacting particle systems.
Tue, 03/22/2011 - 16:01 — SomNambulAuthors: Milton Jara, Patricia Gonçalves
Subjects: Probability
AbstractUsing the renormalization method introduced in \cite{GJ}, we prove what we
call the {\em local} Boltzmann-Gibbs principle for conservative, stationary
interacting particle systems in dimension $d=1$. As applications of this
result, we obtain various scaling limits of additive functionals of particle
systems, like the occupation time of a given site or extensive additive fields
of the dynamics. As a by-product of these results, we also construct a novel
process, related to the stationary solution of the stochastic Burgers equation.Crossover to the KPZ equation.
Mon, 04/19/2010 - 15:01 — SomNambulAuthors: Milton Jara, Patrícia Gonçalves
Subjects: Probability
AbstractWe characterize the crossover regime to the KPZ equation for a class of
weakly asymmetric exclusion processes. This crossover depends on the strength
asymmetry $an^{2-\gamma}$ ($a,\gamma>0$) and it occurs at $\gamma=1/2$. We show
that the density field is governed by an Ornstein-Uhlenbeck if
$\gamma\in(1/2,1)$, while for $\gamma=1/2$ it is an energy solution of the KPZ
equation. The corresponding crossover for the current of particles is readily
obtained.Limit theorems for additive functionals of a Markov chain.
Wed, 12/16/2009 - 16:01 — SomNambulAuthors: Milton Jara, Tomasz Komorowski, Stefano Olla
Subjects: Probability
AbstractConsider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability
measure $\pi$. Let $\Psi$ a function on the state space of the chain, with
$\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient
conditions on the probability transition to prove convergence in law of
$N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. ``Martingale
approximation'' approach and ``coupling'' approach give two different sets of
conditions. We extend these results to continuous time Markov jump processes
$X_t$, whose skeleton chain satisfies our assumptions.Totally Asymmetric Zero-Range process in the Rarefaction Fan.
Fri, 12/04/2009 - 16:01 — SomNambulAuthors: Milton Jara, Patricia Goncalves
Subjects: Probability
AbstractWe consider the one-dimensional totally asymmetric zero-range starting from a
step decreasing profile leading in the hydrodynamic limit to the rarefaction
fan of the associated hydrodynamic equation. We show that the sum of joint
probabilities for second class particles sharing the same site, is convergent
and we compute its limit.Hydrodynamic limit of the exclusion process in inhomogeneous media.
Mon, 08/31/2009 - 17:16 — SomNambulAuthors: Milton Jara
Subjects: Probability
AbstractWe obtain the hydrodynamic limit of a simple exclusion process in an
inhomogeneous environment of divergence form. Our main assumption is a suitable
version of Gamma-convergence for the environment. In this way we obtain an
unified approach to recent works on the field.
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