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Milton Jara

Publications and source records attributed to Milton Jara.

At least 37 records · Page 2Linked to original sources
arXiv2019

Stationary Directed Polymers and Energy Solutions of the Burgers Equation

We consider the stationary O'Connell-Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.

math.PR↗
arXiv2018

Spectral Gap Inequality for Long-Range Random Walks

We show that the spectral gap of a random walk on the domain of normal attraction of an $α$-stable law is of order $\mathcal O(n^α)$ when restricted to boxes of size $n$. The proof is based on a comparison principle that may be of independent interest. The comparison principle also allows to derive a sharp bound on the spectral gap of exclusion and zero-range processes with long jumps when restricted to finite boxes in terms of the gap on the complete graph.

math.PR↗
arXiv2018

Scaling of the Sasamoto-Spohn model in equilibrium

We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of \cite{GJS} and does not use any spectral gap argument.

math.PR↗
arXiv2018

Non-equilibrium Fluctuations of Interacting Particle Systems

We obtain the large scale limit of the fluctuations around its hydrodynamic limit of the density of particles of a weakly asymmetric exclusion process in dimension up to three. The proof is based upon a sharp estimate on the relative entropy of the law of the process with respect to product reference measures associated to the hydrodynamic limit profile, which holds in any dimension and is of independent interest. As a corollary of this entropy estimate, we obtain some quantitative bounds on the speed of convergence of the aforementioned hydrodynamic limit.

math.PR↗
arXiv2018

Non-equilibrium and stationary fluctuations for the SSEP with slow boundary

We derive the non-equilibrium fluctuations of one-dimensional symmetric simple exclusion processes in contact with slowed stochastic reservoirs which are regulated by a factor $n^{-θ}$. Depending on the range of $θ$ we obtain processes with various boundary conditions. Moreover, as a consequence of the previous result we deduce the non-equilibrium stationary fluctuations by using the matrix ansatz method which gives us information on the stationary measure for the model. The main ingredient to prove these results is the derivation of precise bounds on the two-point space-time correlation function, which are a consequence of precise bounds on the transition probability of some underlying random walks.

math.PR↗
arXiv2018

Non-equilibrium fluctuations for a reaction-diffusion model via relative entropy

We look at a superposition of symmetric simple exclusion and Glauber dynamics in the discrete torus in dimension 1. For this model, we prove that the fluctuations around the hydrodynamic limit are described, in the diffusive scale, by an infinite-dimensional Ornstein-Uhlenbeck process. Our proof technique is an adaptation of Yau's Relative Entropy Method that is robust enough to be adapted to other exclusion models. To cut the technical details to a minimum, we assume that the process starts from a product measure with a custom-chosen density, for which the solution of the hydrodynamic equation is stationary. Although we prove fluctuations only in dimension 1, we provide an estimate on the entropy production that holds for any dimension and a proof of the Boltzmann-Gibbs principle that applies in dimension smaller than 3.

math.PR↗
arXiv2018

Symmetric exclusion as a random environment: invariance principle

We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes, a Brownian motion and a Gaussian process with stationary increments.

math.PR↗
arXiv2018

Scaling limit for a long-range divisible sandpile

We study the scaling limit of a divisible sandpile model associated to a truncated $α$-stable random walk. We prove that the limiting distribution is related to an obstacle problem for a truncated fractional Laplacian. We also provide, as a fundamental tool, precise asymptotic expansions for the corresponding rescaled discrete Green's functions. In particular, the convergence rate of these Green's functions to its continuous counterpart is derived.

math.AP↗
arXiv2017

Density fluctuations for exclusion processes with long jumps

We show that the stationary density fluctuations of exclusion processes with long jumps, whose rates are of the form $c^\pm |y-x|^{-(1+α)}$ where $c\pm$ depends on the sign of $y-x$, are given by a fractional Ornstein-Uhlenbeck process for $α\in (0,\frac{3}{2})$. When $α=\frac{3}{2}$ we show that the density fluctuations are tight, in a suitable topology, and that any limit point is an energy solution of the fractional Burgers equation, previously introduced in \cite{GubJar} in the finite volume setting.

math.PR↗
arXiv2017

Second order Boltzmann-Gibbs principle for polynomial functions and applications

In this paper we give a new proof of the second order Boltzmann-Gibbs principle. The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards 1) a trivial process in case of supper-diffusive systems, 2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.

math.PR↗
arXiv2017

Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field Model: Universality Persistence

In [2] it has been proved that a linear Hamiltonian lattice field perturbed by a conservative stochastic noise belongs to the 3/2-Lévy/Diffusive universality class in the nonlinear fluctuating theory terminology [15], i.e. energy superdiffuses like an asymmetric stable 3/2-Lévy process and volume like a Brownian motion. According to this theory this should remain valid at zero tension if the harmonic potential is replaced by an even potential. In this work we consider a quartic anharmonicity and show that the result obtained in the harmonic case persists up to some small critical value of the anharmonicity.

math-ph↗
arXiv2017

Interpolation process between standard diffusion and fractional diffusion

We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume [5, 3]. We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper we investigate the nature of the energy fluctuations for a critical value of the intensity. We show that the latter are described by an Ornstein-Uhlenbeck process driven by a Lévy process which interpolates between Brownian motion and the maximally asymmetric 3/2-stable Lévy process. This result extends and solves a problem left open in [4].

math.PR↗
arXiv2017

Lattice model for Fast Diffusion Equation

We obtain a fast diffusion equation (FDE) as scaling limit of a sequence of zero-range process with symmetric unit rate. Fast diffusion effect comes from the fact that the diffusion coefficient goes to infinity as the density goes to zero. Therefore, in order to capture the behaviour for an arbitrary small density of particles, we consider a proper rescaling of a model with a typically high number of particles per site. Furthermore, we obtain some results on the convergence for the method of lines for FDE.

math.PR↗
arXiv2016

Weakly harmonic oscillators perturbed by a conservative noise

We consider a chain of weakly harmonic coupled oscillators perturbed by a conservative noise. We show that by tuning accordingly the coupling constant energy can diffuse like a Brownian motion or superdiffuse like a maximally 3/2-stable asymmetric L{é}vy process. For a critical value of the coupling, the energy diffusion is described by a family of L{é}vy processes which interpolates between these two processes.

math.PR↗
arXiv2016

Stochastic Burgers equation from long range exclusion interactions

We consider one-dimensional exclusion processes with long jumps given by a transition probability of the form $p_n(\cdot)=s(\cdot)+γ_na(\cdot)$, such that its symmetric part $s(\cdot)$ is irreducible with finite variance and its antisymmetric part is absolutely bounded by $s(\cdot).$ We prove that under diffusive time scaling and strength of asymmetry $\sqrt n γ_n \to_{n\to\infty} b\neq 0$, the equilibrium density fluctuations are given by the unique energy solution of the stochastic Burgers equation.

math.PR↗
arXiv2016

Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment

We consider a system of independent random walks in a common random environment. Previously, a hydrodynamic limit for the system of RWRE was proved under the assumption that the random walks were transient with positive speed. In this paper we instead consider the case where the random walks are transient but with a sublinear speed of the order $n^κ$ for some $κ\in (0,1)$ and prove a quenched hydrodynamic limit for the system of random walks with time scaled by $n^{1/κ}$ and space scaled by $n$. The most interesting feature of the hydrodynamic limit is that the influence of the environment does not average out under the hydrodynamic scaling; that is, the asymptotic particle density depends on the specific environment chosen. The hydrodynamic limit for the system of RWRE is obtained by first proving a hydrodynamic limit for a system of independent particles in a directed trap environment.

math.PR↗
arXiv2016

Explicit LDP for a slowed RW driven by a symmetric exclusion process

We consider a random walk (RW) driven by a simple symmetric exclusion process (SSE). Rescaling the RW and the SSE in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle. The corresponding large deviation rate function can be split into two components, the rate function of the SSE and the one of the RW given the path of the SSE. Such components have different structures (Gaussian and Poissoinian, respectively) and to overcome this difficulty we make use of the theory of Orlicz spaces. In particular, the component of the rate function corresponding to the RW is explicit.

math.PR↗