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Tertuliano Franco

Publications and source records attributed to Tertuliano Franco.

At least 19 recordsLinked to original sources
arXiv2026

Finite reservoirs lead to Wentzell boundary conditions for independent random walks and exclusion process

We analyze the scaling limits (hydrodynamics/propagation of local equilibrium) of two particle systems, namely independent random walks (IRW) and symmetric exclusion processes (SEP), both in the discrete one-dimensional segment where the left boundary is in contact with a reservoir, which may hold any number of particles. At rate one a particle jumps from site $1$ to the reservoir, and at rate $αη(0) N^{-θ}$ a particle jumps from the reservoir to site $1$ (if site $1$ is empty in SEP), where $η(0)$ is the total number of particles in the reservoir and $θ\geq 0$ is a parameter whose tuning leads to a dynamical phase transition. For every $θ\geq 0$, the hydrodynamic equation is the heat equation with Neumann boundary conditions at the right boundary for both processes, while the boundary conditions at the left boundary depend on the value of $θ$. For $0\leq θ<1$, it is given by Neumann boundary conditions, meaning the deposit is asymptotically empty, acting as a barrier. For $θ>1$, and for IRW (resp. SEP) it is given by a non-homogeneous (resp. homogeneous) Dirichlet boundary condition, meaning the reservoir becomes asymptotically infinite, acting as a heat bath (resp. the reservoir behaves as a sink). For $θ=1$, we obtain a non-local Dirichlet boundary condition, which is additionally non-linear for SEP. As a by-product, we obtain an equivalence between solutions to the heat equation with Wentzell boundary conditions and with non-local Dirichlet boundary conditions related to the total mass of the system.

math.PR↗
arXiv2026

A Functional Central Limit Theorem for the General Brownian Motion on the Half-Line

In this work, we establish a Trotter-Kato type theorem. More precisely, we characterize the convergence in distribution of Feller processes by examining the convergence of their generators. The main novelty lies in providing quantitative estimates in the vague topology at any fixed time. As important applications, we deduce functional central limit theorems for random walks on the positive integers with boundary conditions, which converge to Brownian motions on the positive half-line with boundary conditions at zero.

math.PR↗
arXiv2026

Delayed logistic equation as a limit of long memory Markov chains

We introduce and analyze a long-memory continuous-time Markov chain on $\mathbb{R}_{+}$ whose jump mechanism depends explicitly on a state in the past. From the present state $x_0$, the process jumps to $x_0\left(1+\frac{1}{N}\right)$ or $x_0\left(1-\frac{x_{-\lfloor τN \rfloor}}{N^2}\right)$, each at rate $\tfrac{1}{2}$, where $x_{-\lfloor τN \rfloor}$ denotes the state located $\lfloor τN \rfloor$ jumps backward in time. Here the delay $τ> 0$ is fixed and $N$ is the scaling parameter. The initial condition is prescribed by a vector of length $\lfloor τN \rfloor + 1$, all of whose entries are equal to $μN$. Using a genuine space-time replacement lemma, we prove that, as $N \to \infty$, the rescaled process converges to a deterministic limit governed by the Delayed Logistic Equation (also known as the Hutchinson equation) with delay $τ$ and initial condition $ρ(t) \equiv μ$ for $t \in [-τ, 0]$.

math.PR↗
arXiv2025

The Most General Brownian Motion on the Line and on Two Closed Half-Lines

In the 1950s, W. Feller characterized the most general Brownian motion on the closed half-line. He showed that any such process is a mixture of reflected, sticky, and killed Brownian motions. By most general Brownian motion, we mean a strong Markov process whose excursions away from zero coincide with those of standard Brownian motion, and which may be sent to the cemetery state upon hitting zero. In this work, we fully characterize the most general Brownian motion on the whole real line and on the union of two closed half-lines. Our results are twofold. First, we show that the most general Brownian motion on the line is the process known in the literature as the Skew Sticky Brownian Motion Killed at Zero (see Borodin and Salminen's book). Second, we prove that the most general Brownian motion on two closed half-lines is a process, which we call the Skew Sticky Killed at Zero Snapping Out Brownian Motion. This process extends the Snapping Out Brownian Motion introduced by A. Lejay in 2016.

math.PR↗
arXiv2025

Hydrodynamic limit for repeated averages on the complete graph

We establish a hydrodynamical limit for the averaging process on the complete graph with N vertices, showing that, after a timescale of order N, the empirical distribution of opinions converges to a unique measure. Moreover, if the initial distribution is absolutely continuous concerning the Lebesgue measure, the limiting measure remains absolutely continuous and its density satisfies a non-diffusive differential equation, that resembles the Smoluchowski coagulation equation.

math.PR↗
arXiv2024

Superdiffusive Scaling Limits for the Symmetric Exclusion Process with Slow Bonds

In \cite{fgn1}, the hydrodynamic limit in the diffusive scaling of the symmetric simple exclusion process with a finite number of slow bonds of strength $n^{-β}$ has been studied. Here $n$ is the scaling parameter and $β>0$ is fixed. As shown in \cite{fgn1}, when $β>1$, such a limit is given by the heat equation with Neumann boundary conditions. In this work, we find more non-trivial super-diffusive scaling limits for this dynamics. Assume that there are $k$ equally spaced slow bonds in the system. If $k$ is fixed and the time scale is $k^2n^θ$, with $θ\in (2,1+β)$, the density is asymptotically constant in each of the $k$ boxes, and equal to the initial expected mass in that box, i.e., there is no time evolution. If $k$ is fixed and the time scale is $k^2n^{1+β}$, then the density is also spatially constant in each box, but evolves in time according to the discrete heat equation. Finally, if the time scale is $k^2n^{1+β}$ and, additionally, the number of boxes $k$ increases to infinity, then the system converges to the continuous heat equation on the torus, with no boundary conditions.

math.PR↗
arXiv2024

A Central Limit Theorem for intransitive dice

Intransitive dice $D^{(1)}, \ldots, D^{(\ell)}$ are dice such that $D^{(1)}$ has advantage when played against $D^{(2)}$, dice $D^{(2)}$ has advantage when played against $D^{(3)}$ and so on, up to $D^{(\ell)}$, which has advantage over $D^{(1)}$. In this twofold work, we first present (deterministic) results on the existence of general intransitive dice. Second and mainly, a central limit theorem for the vector of normalized victories of a die against the next one in the list when the faces of a die are i.i.d.\ random variables and all dice are independent, but different dice may have distinct distributions associated with them, as well as they may have distinct numbers of faces. Exploiting this central limit theorem, we derive two major consequences. First, we are able to obtain first order exponential asymptotics for the number of $\ell$-tuples of intransitive dice, when the number of faces of the dice grows. Second, we obtain a criterion to ensure that the asymptotic probability of observing intransitive dice is null, which applies to many cases, including all continuous distributions and many discrete ones.

math.PR↗
arXiv2024

A strong large deviation principle for the empirical measure of random walks

In this article we show that the empirical measure of certain continuous time random walks satisfies a strong large deviation principle with respect to a topology introduced in~\cite{MV2016} by Mukherjee and Varadhan. This topology is natural in models which exhibit an invariance with respect to spatial translations. Our result applies in particular to the case of simple random walk and complements the results obtained in~\cite{MV2016} in which the large deviation principle has been established for the empirical measure of Brownian motion.

math.PR↗
arXiv2023

Nonequilibrium Joint Fluctuations for Current and Occupation Time in The Symmetric Exclusion Process

We provide a full description for the joint fluctuations of current and occupation time in the one-dimensional nonequilibrium simple symmetric exclusion process, furnishing explicit formulas for the covariances of the limiting Gaussian process. The main novelties consist of a proof of the tightness of the nonequilibrium current based on new correlation estimates, refined estimates on the discrete gradient of the transition probabilities of the SSEP, and a nonequilibrium Kipnis-Varadhan Lemma based on a Fourier approach.

math.PR↗
arXiv2021

Large Deviations for the SSEP with slow boundary: the non-critical case

We prove a large deviations principle for the empirical measure of the one dimensional symmetric simple exclusion process in contact with reservoirs. The dynamics of the reservoirs is slowed down with respect to the dynamics of the system, that is, the rate at which the system exchanges particles with the boundary reservoirs is of order $n^{-θ}$, where $n$ is number of sites in the system, $θ$ is a non negative parameter, and the system is taken in the diffusive time scaling. Two regimes are studied here, the subcritical $θ\in(0,1)$ whose hydrodynamic equation is the heat equation with Dirichlet boundary conditions and the supercritical $θ\in(1,+\infty)$ whose hydrodynamic equation is the heat equation with Neumann boundary conditions. In the subcritical case $θ\in(0,1)$, the rate function that we obtain matches the rate function corresponding to the case $θ=0$ which was derived on previous works (see \cite{blm,flm}), but the challenges we faced here are much trickier. In the supercritical case $θ\in(1,+\infty)$, the rate function is equal to infinity outside the set of trajectories which preserve the total mass, meaning that, despite the discrete system exchanges particles with the reservoirs, this phenomena has super-exponentially small probability in the diffusive scaling limit.

math.PR↗
arXiv2021

Large Deviations in the Supremum Norm for a Reaction-Diffusion System

We present large deviations estimates in the supremum norm for a system of independent random walks superposed with a birth-and-death dynamics evolving on the discrete torus with $N$ sites. The scaling limit considered is the so-called \textit{high density limit} (see the survey \cite{franco} on the subject), where space, time and initial quantity of particles are rescaled. The associated rate functional here obtained is a semi-linearised version of the rate function of \cite{JonaLandimVares}, which dealt with large deviations of exclusion processes superposed with birth-and-death dynamics. An ingredient in the proof of large deviations consists in providing a limit of a suitable class of perturbations of the original process. This is precisely one of the main contributions of this work: a strategy to extend the original high density approach (as in \cite{Arnold,blount2,blount,francogroisman,Kote2,KoteHigh1988}) to weakly asymmetric systems. Two cases are considered with respect to the initial quantity of particles, the power law and the (at least) exponential growth. In the first case, we present the lower bound only on a certain subset of smooth profiles, while in the second case, additionally assuming concavity of the birth and the death functions and a constant initial profile, we provide a full large deviations principle.

math.PR↗
arXiv2020

The Directed Edge Reinforced Random Walk: The Ant Mill Phenomenon

We define here a \textit{directed edge reinforced random walk} on a connected locally finite graph. As the name suggests, this walk keeps track of its past, and gives a bias towards directed edges previously crossed proportional to the exponential of the number of crossings. The model is inspired by the so called \textit{Ant Mill phenomenon}, in which a group of army ants forms a continuously rotating circle until they die of exhaustion. For that reason we refer to the walk defined in this work as the \textit{Ant RW}. Our main result justifies this name. Namely, we will show that on any finite graph which is not a tree, and on $\mathbb Z^d$ with $d\geq 2$, the Ant RW almost surely gets eventually trapped into some directed circuit which will be followed forever. In the case of~$\mathbb Z$ we show that the Ant RW eventually escapes to infinity and satisfies a law of large number with a random limit which we explicitly identify.

math.PR↗
arXiv2019

The Slow Bond Random Walk and the Snapping Out Brownian Motion

We consider the continuous time symmetric random walk with a slow bond on $\mathbb Z$, which rates are equal to $1/2$ for all bonds, except for the bond of vertices $\{-1,0\}$, which associated rate is given by $αn^{-β}/2$, where $α\geq 0$ and $β\in [0,\infty]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $β<1$, then it converges to the usual Brownian motion. If $β\in (1,\infty]$, then it converges to the reflected Brownian motion. And at the critical value $β=1$, it converges to the snapping out Brownian motion (SNOB) of parameter $κ=2α$, which is a Brownian type-process recently constructed in Lejay, A., The snapping out Brownian motion. Ann. Appl. Probab., 26(3):1727--1742, 2016. We also provide Berry-Esseen estimates in the dual bounded Lipschitz metric for the weak convergence of one-dimensional distributions, which we believe to be sharp.

math.PR↗
arXiv2018

Hydrodynamic Limit for the SSEP with a Slow Membrane

In this paper we consider a symmetric simple exclusion process (SSEP) on the $d$-dimensional discrete torus $\mathbb{T}^d_N$ with a spatial non-homogeneity given by a slow membrane. The slow membrane is defined here as the boundary of a smooth simple connected region $Λ$ on the continuous $d$-dimensional torus $\mathbb{T}^d$. In this setting, bonds crossing the membrane have jump rate $α/N^β$ and all other bonds have jump rate one, where $α>0$, $β\in[0,\infty]$, and $N\in \mathbb{N}$ is the scaling parameter. In the diffusive scaling we prove that the hydrodynamic limit presents a dynamical phase transition, that is, it depends on the regime of $β$. For $β\in[0,1)$, the hydrodynamic equation is given by the usual heat equation on the continuous torus, meaning that the slow membrane has no effect in the limit. For $β\in(1,\infty]$, the hydrodynamic equation is the heat equation with Neumann boundary conditions, meaning that the slow membrane $\partial Λ$ divides $\mathbb{T}^d$ into two isolated regions $Λ$ and $Λ^\complement$. And for the critical value $β=1$, the hydrodynamic equation is the heat equation with certain Robin boundary conditions related to the Fick's Law.

math.PR↗
arXiv2018

Non-equilibrium fluctuations for the SSEP with a slow bond

We prove the non-equilibrium fluctuations for the one-dimensional symmetric simple exclusion process with a slow bond. This generalizes a result of T. Franco, A. Neumann and P. Gonçalves (2013), which dealt with the equilibrium fluctuations. The foundation stone of our proof is a precise estimate on the correlations of the system, and that is by itself one of the main novelties of this paper. To obtain these estimates, we first deduce a spatially discrete PDE for the covariance function and we relate it to the local times of a random walk in a non-homogeneous environment via Duhamel's principle. Projection techniques and coupling arguments reduce the analysis to the problem of studying the local times of the classical random walk. We think that the method developed here can be applied to a variety of models, and we provide a discussion on this matter.

math.PR↗
arXiv2017

Crossover to the stochastic Burgers equation for the WASEP with a slow bond

We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $ρ\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^β}2}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\frac1{\vphantom{n^β}2}-\frac{a}{2n^{\vphantomβγ}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\fracα{2n^β}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\fracα{2n^β}-\frac{a}{2n^{\vphantomβγ}}$). Above, $α>0$, $γ\geq β\geq 0$, $a\geq 0$. For $β<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $γ>\frac12$, while for $γ= \frac12$ it is an energy solution of the stochastic Burgers equation. For $γ\geqβ=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $γ\geqβ> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.

math.PR↗
arXiv2017

Non-equilibrium and stationary fluctuations of a slowed boundary symmetric exclusion

We consider a one-dimensional symmetric simple exclusion process in contact with slowed reservoirs: at the left (resp. right) boundary, particles are either created or removed at rates given by $α/n$ or $(1-α)/n$ (resp. $β/n$ or $(1-β)/n$) where $α, β>0$ and $n$ is a scaling parameter. We obtain the non-equilibrium fluctuations and consequently the non-equilibrium stationary fluctuations.

math.PR↗