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Tertuliano Franco

Publications and source records attributed to Tertuliano Franco.

35 records · Page 2Linked to original sources
arXiv2017

Large deviations for the exclusion process with a slow bond

We consider the one-dimensional symmetric simple exclusion process with a slow bond. In this model, whilst all the transition rates are equal to one, a particular bond, the \emph{slow bond}, has associated transition rate of value $N^{-1}$, where $N$ is the scaling parameter. This model has been considered in previous works on the subject of hydrodynamic limit and fluctuations. In this paper, assuming uniqueness for weak solutions of hydrodynamic equation associated to the perturbed process, we obtain dynamical large deviations estimates in the diffusive scaling. The main challenge here is the fact that the presence of the slow bond gives rise to Robin's boundary conditions in the \emph{continuum}, substantially complicating the large deviations scenario.

math.PR↗
arXiv2016

Phase transition in the KMP model with Slow/Fast boundaries

The Kipnis-Marchioro-Presutti (KMP) is a known model consisting on a one-dimensional chain of mechanically uncoupled oscillators, whose interactions occur via independent Poisson clocks: when a Poisson clock rings, the total energy at two neighbors is redistributed uniformly at random between them. Moreover, at the boundaries, energy is exchanged with reservoirs of fixed temperatures. We study here a generalization of the KMP model by considering different rates at energy is exchanged with the reservoirs, and we then prove the existence of a phase transition for the heat flow.

math.PR↗
arXiv2016

Equilibrium fluctuations for the slow boundary exclusion process

We prove that the equilibrium fluctuations of the symmetric simple exclusion process in contact with slow boundaries is given by an Ornstein-Uhlenbeck process with Dirichlet, Robin or Neumann boundary conditions depending on the range of the parameter that rules the slowness of the boundaries.

math.PR↗
arXiv2016

Corrigendum to: Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We present the correct space of test functions for the Ornstein-Uhlenbeck processes defined in \cite{fgn2}. Under these new spaces, an invariance with respect to a second order operator is shown, granting the existence and uniqueness of those processes. Moreover, we detail how to prove some properties of the semi-groups, which are required in the proof of uniqueness.

math.PR↗
arXiv2015

Scalings limits for the exclusion process with a slow site

We consider the symmetric simple exclusion processes with a slow site in the discrete torus with $n$ sites. In this model, particles perform nearest-neighbor symmetric random walks with jump rates everywhere equal to one, except at one particular site, \textit{the slow site}, where the jump rate of entering that site is equal to one, but the jump rate of leaving that site is given by a parameter $g(n)$. Two cases are treated, namely $g(n)=1+o(1)$, and $g(n)=αn^{-β}$ with $β>1$, $α>0$. In the former, both the hydrodynamic behavior and equilibrium fluctuations are driven by the heat equation (with periodic boundary conditions when in finite volume). In the latter, they are driven by the heat equation with Neumann boundary conditions. We therefore establish the existence of a dynamical phase transition. The critical behavior remains open.

math.PR↗
arXiv2014

Interacting particle systems: hydrodynamic limit versus high density limit

In this short survey we compare aspects of two different approaches for scaling limits of interacting particle systems, the hydrodynamic limit and the high density limit. We present some examples, comments and open problems on each approach for different scenarios: the law of large numbers, the central limit theorem, and the large deviations principle. It is given a special attention to a recent result of Franco and Groisman' 12 about the high density limit of a system exhibiting explosion of the total numbers of particles in finite time.

math.PR↗
arXiv2014

Symmetric exclusion as a random environment: hydrodynamic limits

We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly non-uniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for the exclusion as seen from the walk without explicit knowledge of its invariant measures. We further discuss how to obtain similar results for several variants of this model.

math.PR↗
arXiv2013

Continuous dependence on the derivative of generalized heat equations

We consider here a generalized heat equation $\partial_t ρ=\frac{d}{dx}\frac{d}{dW}ρ$, where $W$ is a finite measure on the one dimensional torus, and $\frac{d}{dW}$ is the Radon-Nikodym derivative with respect to $W$. Such equation has appeared in different contexts, being related to physical systems and representing a large class of classical and non-classical parabolic equations. As a natural assumption on $W$, we require that the Lebesgue measure is absolutely continuous with respect to $W$. The main result here presented consists in proving, for a suitable topology, a continuous dependence of the solution $ρ$ as a function of $W$.

math.AP↗
arXiv2013

Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We analyze the equilibrium fluctuations of the density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is $αn^-β$, where $α,β\geq{0}$ and $n$ is the scaling parameter. Depending on the regime of $β$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value $β=1$, starting a tagged particle near the slow bond, we obtain a family of gaussian processes indexed in $α$, interpolating a fractional brownian motion of Hurst exponent 1/4 and the degenerate process equal to zero.

math.PR↗
arXiv2013

Occupation times of exclusion processes with conductances

We obtain the fluctuations for the occupation time of one-dimensional symmetric exclusion processes with speed change, where the transition rates (conductances) are driven by a general function W. The approach does not require sharp bounds on the spectral gap of the system nor the jump rates to be bounded from above or below. We present some examples and for one of them, we observe that the fluctuations of the current are trivial, but the fluctuations of the occupation time are given by a fractional Brownian Motion. This shows that, in general, the fluctuations of the current and of the occupation time are not of same order.

math.PR↗
arXiv2013

Slowed exclusion process: hydrodynamics, fluctuations and phase transitions

This is a short survey on recent results obtained by the authors on dynamical phase transitions of interacting particle systems. We consider particle systems with exclusion dynamics, but it is conjectured that our results should hold for a general class of particle systems. The parameter giving rise to the phase transition is the "slowness" of a single bond in the discrete lattice. The phase transition is verified not only in the hydrodynamics, but also in the fluctuations of the density, the current and the tagged particle. Moreover, we found a phase transition in the continuum, that is, at the level of the hydrodynamic equations, in agreement with the dynamical phase transition for the particle systems.

math.PR↗
arXiv2013

Phase transition of a Heat equation with Robin's boundary conditions and exclusion process

For a heat equation with Robin's boundary conditions which depends on a parameter $α>0$, we prove that its unique weak solution $ρ^α$ converges, when $α$ goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann's boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of $ρ^α$ obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.

math.PR↗
arXiv2012

A particle system with explosions: law of large numbers for the density of particles and the blow-up time

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a stong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion time.

math.PR↗
arXiv2011

Hydrodynamical behavior of symmetric exclusion with slow bonds

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-β}$, with $β\in[0,\infty)$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$. If $β\in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $β=1$, it is given by a parabolic equation involving an operator $\frac{d}{dx}\frac{d}{dW}$, where $W$ is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If $β\in(1,\infty)$, it is given by the heat equation with Neumann's boundary conditions, meaning no passage through the slow bonds in the continuum.

math.PR↗
arXiv2010

Hydrodynamic Limit for a type of Exclusion Processes with slow bonds in dimension $\ge 2$

Let $Λ$ be a connected closed region with smooth boundary contained in the $d$-dimensional continuous torus $\bb T^d$. In the discrete torus $N^{-1} \bb T^d_N$, we consider a nearest neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on $Λ$ in the following way: if both sites are in $Λ$ or $Λ^\complement$, the exchange rate is one; If one site is in $Λ$ and the other one is in $Λ^\complement$ and the direction of the bond connecting the sites is $e_j$, then the exchange rate is defined as $N^{-1}$ times the absolute value of the inner product between $e_j$ and the normal exterior vector to $\pΛ$. We show that this exclusion type process has a non-trivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by $\partialΛ$. Thus the model represents a system of particles under hard core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementar regions.

math.PR↗
arXiv2008

Hydrodynamic limit of gradient exclusion processes with conductances

Fix a strictly increasing right continuous with left limits function $W: \bb R \to \bb R$ and a smooth function $Φ: [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b \le Φ'\le b^{-1}$. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes, with conductances given by $W$, is described by the weak solutions of the non-linear differential equation $\partial_t ρ= (d/dx)(d/dW) Φ(ρ)$. We derive some properties of the operator $(d/dx)(d/dW)$ and prove uniqueness of weak solutions of the previous non-linear differential equation.

math.PR↗