arXiv subjects

Claudio Landim

Publications and source records attributed to Claudio Landim.

At least 19 recordsLinked to original sources

arXiv2026

Full $\Gamma-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions

Consider the diffusion process \begin{equation*} dX_{\epsilon}(t) = \mss b(X_{\epsilon}(t)) \, dt + \sqrt{2\, \epsilon\, \mss a(X_\epsilon(t))} \, dW_{t}, \end{equation*} on the one-dimensional torus $\bb T = [0,1)$. Here $\epsilon$ is the temperature, $W_{t}$ a Brownian motion on $\bb T$ and $\mss a$, $\mss b$ functions of class $C^{2}(\bb T)$ satisfying further conditions. Denote by $\mss P(\bb T)$ the set of probability measures on $\bb T$ equipped with the weak topology, and by $\ms I_{\epsilon}\colon \mss P(\bb T)\to [0,+\infty)$ the level two large deviation rate functional of the diffusion $X_{\epsilon}(\cdot)$. We derive a full $\Gamma-$expansion of $\ms I_{\epsilon}$, as $\epsilon \to 0$, expressing it as \begin{equation*} \ms I_{\epsilon} = \frac{1}{\epsilon} \;\ms J^{(-1)} \; +\; \ms J^{(0)} \;+\; \sum_{p=1}^{\widehat{\mf q}}\frac{1}{\theta^{(p)}_{\epsilon}}\;\ms J^{(p)}\,, \end{equation*} where $\ms J^{(-1)}$, $\ms J^{(0)}$, $\ms J^{(p)} \colon \mss P(\bb T)\to [0,+\infty]$ represent rate functionals, independent of $\epsilon$, and $\theta^{(p)}_{\epsilon}$ are the time-scales at which the Markov process $X_{\epsilon}(\cdot)$ exhibits a metastable behaviour.

math.PR↗

arXiv2026

Convergence of the Condensing Symmetric Inclusion Process on the Torus in the Thermodynamical Limit to Coalescing Brownian Motions

We investigate the saturation regime of the condensing symmetric inclusion process on the discrete one-dimensional torus in the thermodynamical limit. In this regime, the total mass concentrates on a finite number of sites, forming condensates. Our main result establishes that, under appropriate scaling, the positions of the condensates converge to a system of coalescing Brownian motions on the continuum torus. In particular, condensates perform diffusive motion until they meet, at which point they merge and their masses coagulate. This provides a rigorous derivation of a macroscopic coalescing diffusion from an underlying interacting particle system with condensation. The main technical difficulty arises from the complicated coalescence mechanism of two condensates of particles, whose trajectories are very difficult to track completely. The key idea is to control the coalescing time instead and prove that it is negligible compared to the time-scale of condensate movement. By combining this with precise estimates of movements without coalescence, we can prove its convergence to coalescing Brownian motions.

math.PR↗

arXiv2026

Critical stationary fluctuations in reaction--diffusion processes

We study stationary fluctuations at criticality for a one-dimensional reaction--diffusion process combining symmetric simple exclusion dynamics with Glauber-type spin flips. The strength of the Glauber interaction is tuned to the critical regime in which the quadratic term in the effective potential vanishes. Focusing on the stationary distribution, we show that the total magnetization scaled by $n^{3/4}$ exhibits non-Gaussian fluctuations. More precisely, we prove that under the invariant measure the rescaled magnetization converges in distribution to a random variable with density proportional to $\exp\{-2(\theta y^2 + y^4/2)\}$. In contrast with the previous result, we show that the density field acting on the faster modes, that is, those associated to zero-mean test functions, have much smaller Gaussian fluctuations. It follows from the previous two results that the rescaled density field projects onto the magnetization in the sense that its action on zero-mean test functions vanishes in the limit.

math.PR↗

arXiv2026

Dimension-decaying diffusion processes as the scaling limit of condensing zero-range processes

In this article, we prove that, on the diffusive time scale, condensing zero-range processes converge to a dimension-decaying diffusion process on the simplex \[ \Sigma = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in S} x_i = 1\}, \] where $S$ is a finite set. This limiting diffusion has the distinctive feature of being absorbed at the boundary of the simplex. More precisely, once the process reaches a face \[ \Sigma_A = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in A} x_i = 1\}, \qquad A \subset S, \] it remains confined to this set and evolves in the corresponding lower-dimensional simplex according to a new diffusion whose parameters depend on the subset $A$. This mechanism repeats itself, leading to successive reductions of the dimension, until one of the vertices of the simplex is reached in finite time. At that point, the process becomes permanently trapped. The proof relies on a method to extend the domain of the associated martingale problem, which may be of independent interest and useful in other contexts.

math.PR↗

arXiv2025

The Gamma Expansion of the Level Two Large Deviation Rate Functional for Reversible Diffusion Processes

Fix a smooth Morse function $U\colon \mathbb{R}^{d}\to\mathbb{R}$ with finitely many critical points, and consider the solution of the stochastic differential equation \[ d\boldsymbol{x}_{\epsilon}(t)=-\nabla U(\boldsymbol{x}_{\epsilon}(t))\,dt \,+\,\sqrt{2\epsilon}\, d\boldsymbol{w}_{t}\,, \] where $(\boldsymbol{w}_{t})_{t\ge0}$ represents a $d$-dimensional Brownian motion, and $\epsilon>0$ a small parameter. Denote by $\mathcal{P}(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^d$, and by $\mathcal{I}_{\epsilon} \colon \mathcal{P}(\mathbb{R}^{d})\to[0,\,\infty]$ the Donsker--Varadhan level two large deviations rate functional. We express $\mathcal{I}_\epsilon$ as $\mathcal{I}_\epsilon = \epsilon^{-1} \mathcal{J}^{(-1)} + \mathcal{J}^{(0)} + \sum_{1\le p\le \mathfrak{q}} (1/\theta^{(p)}_\epsilon) \, \mathcal{J}^{(p)}$, where $\mathcal{J}^{(p)}\colon \mathcal{P}(\mathbb{R}^d) \to [0,+\infty]$ stand for rate functionals independent of $\epsilon$ and $\theta^{(p)}_\epsilon$ for sequences such that $\theta^{(1)}_\epsilon \to\infty$, $\theta^{(p)}_\epsilon / \theta^{(p+1)}_\epsilon \to 0$ for $1\le p< \mathfrak{q}$. The speeds $\theta^{(p)}_\epsilon$ correspond to the time-scales at which the diffusion $\boldsymbol{x}_{\epsilon}(\cdot)$ exhibits a metastable behaviour, while the functional $\mathcal{J}^{(p)}$ represent the level two, large deviations rate functionals of the finite-state, continuous-time Markov chains which describe the evolution of the diffusion $\boldsymbol{x}_{\epsilon}(\cdot)$ among the wells in the time-scale $\theta^{(p)}_\epsilon$.

math.PR↗

arXiv2025

Critical dynamical fluctuations in reaction-diffusion processes

We consider a one-dimensional microscopic reaction-diffusion process obtained as a superposition of a Glauber and a Kawasaki dynamics. The reaction term is tuned so that a dynamical phase transition occurs in the model as a suitable parameter is varied. We study dynamical fluctuations of the density field at the critical point. We characterise the slowdown of the dynamics at criticality, and prove that this slowdown is induced by a single observable, the global density (or magnetisation). We show that magnetisation fluctuations are non-Gaussian and characterise their limit as the solution of a non-linear SDE. We prove, furthermore, that other observables remain fast: the density field acting on the fast modes (i.e. on mean-0 test functions) and with Gaussian scaling converges, in the sense of finite dimensional distributions, to a Gaussian field with space-time covariance that we compute explicitly. The proof relies on a decoupling of slow and fast modes relying in particular on a relative entropy argument. Major technical difficulties include the fact that local equilibrium does not hold due to the non-linearity, and proving replacement estimates on diverging time intervals due to critical slowdown.

math.PR↗

arXiv2025

From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics

Consider the one-dimensional elliptic operator given by \begin{equation*} (L_\epsilon f)(x) \;=\; b (x) \, f'(x) \,+\, \epsilon\, a (x)\, f''(x) \;, \end{equation*} where the drift $b\colon R \to R$ and the diffusion coefficient $a\colon R \to R$ are periodic $C^1(R)$ functions satisfying further conditions, and $\epsilon>0$. Consider the initial-valued problem \begin{equation*} \left\{ \begin{aligned} & \partial_{t}\,u_{\epsilon}\,=\,L_{\epsilon}\,u_{\epsilon}\;,\\ & u_{\epsilon}(0,\,\cdot)=u_{0}(\cdot)\;, \end{aligned} \right.\end{equation*} for some bounded continuous function $u_{0}$. We prove the existence of time-scales $\theta_{\epsilon}^{(1)},\,\dots,\,\theta_{\epsilon}^{(\mathfrak{q})}$ such that $\theta_{\epsilon}^{(1)}\to\infty$, $\theta_{\epsilon}^{(p+1)}/\theta_{\epsilon}^{(p)}\to\infty$, $1\le p\le\mathfrak{q}-1$, probability measures $p(x,\cdot)$, $x\in R$, and kernels $R_{t}^{(p)}(m_j,m_k)$, where $\{m_j:j\in Z\}$ represents the set of stable equilibrium of the ODE $\dot{x}(t) = b(x(t))$ such that \begin{equation*} \lim_{\epsilon\to0} u_{\epsilon}(t\theta_{\epsilon}^{(p)}, x) \;=\;\sum_{j,k\in Z} p(x,m_j)\, R_{t}^{(p)} (m_j,m_k) \,u_{0}(m_k)\;, \end{equation*} for all $t>0$ and $x\in R$. The solution $u_{\epsilon}$ asymptotic behavior description is completed by the characterisation of its behaviour in the intermediate time-scales $\varrho_{\epsilon}$ such that $\varrho_{\epsilon}/\theta_{\epsilon}^{(p)}\to\infty$, $\varrho_{\epsilon}/\theta_{\epsilon}^{(p+1)}\to0$ for some $0\le p\le\mathfrak{q}$, where $\theta_{\epsilon}^{(0)}=1$, $\theta_{\epsilon}^{(\mathfrak{q}+1)}=+\infty$. The proof relies on the analysis of the diffusion $X_\epsilon(\cdot)$ induced by the generator $L_\epsilon$ based on the resolvent approach to metastability introduced in [21].

math.PR↗

arXiv2025

Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment

We derive a singular diffusion limit for the position of a tagged particle in zero range interacting particle processes on a one dimensional torus with a Sinai-type random environment via two steps. In the first step, a regularization is introduced by averaging the random environment over an $\varepsilon N$-neighborhood. With respect to such an environment, the microscopic drift of the tagged particle is in form $\frac{1}{N}W_\varepsilon'$, where $W_\varepsilon'$ is a regularized White noise. Scaling diffusively, we find the nonequilibrium limit of the tagged particle $x^\varepsilon_t$ is the unique weak solution of $d x_t^{\varepsilon} = 2\frac{\Phi(\rho^{\varepsilon}(t, x_t^{\varepsilon}))}{\rho^{\varepsilon}(t, x_t^\varepsilon)} \,W_{\varepsilon}'(x_t^\varepsilon) + \sqrt{\frac{\Phi(\rho^{\varepsilon}(t, x_t^\varepsilon))}{\rho^{\varepsilon}(t, x_t^\varepsilon)}} \,dB_t$, in terms of the hydrodynamic mass density $\rho^\varepsilon$ recently identified and homogenized interaction rate $\Phi$. In the second step, we show that $x^\varepsilon$, as $\varepsilon$ vanishes, converges in law to the diffusion $x^0$ described informally by $d x_t^0 = 2\frac{\Phi(\rho^{0}(t, x_t^{0}))}{\rho^{0}(t, x_t^0)} \,W'(x_t^0) + \sqrt{\frac{\Phi(\rho^{0}(t, x_t^0))}{\rho^{0}(t, x_t^0)}} \,dB_t$, where $W'$ is a spatial White noise and $\rho^0$ is the para-controlled limit of $\rho^\varepsilon$ also recently identified, solving the singular PDE $ \partial_t \rho^0 = \frac{1}{2}\Delta \Phi(\rho^0) - 2\nabla \big(W' \Phi(\rho^0)\big)$.

math.PR↗

arXiv2024

$\Gamma$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains

Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the measure-current large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = \mathbf I^{(0)} \,+\, \sum_{1\le p\le \mathfrak q} (1/\theta^{(p)}_n) \, \mathbf I^{(p)}$ for some rate functionals $\mathbf I^{(p)}$. The weights $\theta^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ evolves among the metastable wells, and the rate functionals $\mathbf I^{(p)}$ characterise the asymptotic Markovian dynamics among these wells. This expansion provides therefore an alternative description of the metastable behavior of a sequence of Markovian dynamics. Together with the results in \cite{bgl-24,l-gamma}, this work finishes the project of characterising the hierarchical metastable behavior of finite-state Markov chains by means of the $\Gamma$-expansion of large deviations rate functionals. In addition, we present optimal conditions under which the measure (Donsker-Varadhan) or the measure-current large deviations rate functional determines the original dynamics, and calculate the first and second derivatives of the measure large deviations rate functional, thereby generalising the results for i.i.d. random variables.

math.PR↗

arXiv2024

Linear fluctuation of interfaces in Glauber-Kawasaki dynamics

In this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface $\Gamma_t$ formed is stationary that is, $\Gamma_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $\Gamma_t=\Gamma=\{x: x_1=0\}$ is immobile and the hydrodynamic limit is given by $\rho(t,v) = \rho_+$ for $v_1\in (0,1/2)$ and $\rho(t,v)= \rho_-$ for $v_1\in (-1/2,0)$ for all $t\ge 0$, where $v=(v_1,\ldots,v_d)\in T^d$ identified with $[-1/2,1/2)^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N= O(\sqrt{\log(N)})$ in $d\leq 2$. In the one dimensional case, the field limit is given by ${\bf e}(v_1) B_t$ where $B_t$ is a Brownian motion and ${\bf e}$ is the normalized derivative of a decreasing `standing wave' solution $\phi$ of $\partial^2_{v_1} \phi - V'(\phi)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is ${\bf e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function ${\bf e}(\cdot)$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $\phi$.

math.PR↗

arXiv2024

Exclusion processes with non-reversible boundary: hydrodynamics and large deviations

We consider a one-dimensional exclusion dynamics in mild contact with boundary reservoirs. In the diffusive scale, the particles' density evolves as the solution of the heat equation with non-linear Robin boundary conditions. For appropriate choices of the boundary rates, these partial differential equations have more than one stationary solution. We prove the dynamical large deviations principle.

math.PR↗

arXiv2024

Metastability and time scales for parabolic equations with drift 2: the general time scale

Consider the elliptic operator given by \[ \mathscr{L}_\epsilon f=b\cdot\nabla f+\epsilon\Delta f \] for some smooth vector field $b:\mathbb{R}^d\to\mathbb{R}^d$ and $\epsilon>0$, and the initial-valued problem on $\mathbb{R}^d$ \[ \left\{\begin{aligned}&\partial_t u_\epsilon=\mathscr{L}_\epsilon u_\epsilon,\\ &u_\epsilon(0,\,\cdot)=u_0(\cdot), \end{aligned} \right. \] for some bounded continuous function $u_0$. Under the hypothesis that the diffusion on $\mathbb{R}^d$ induced by $\mathscr{L}_\epsilon$ has a Gibbs invariant measure of the form $\exp \{-U(x)/\epsilon\}dx$ for some smooth Morse potential function $U$, we provide the complete characterization of the multi-scale behavior of the solution $u_\epsilon$ in the regime $\epsilon\to0$. More precisely, we find the critical time scales $1\ll \theta_\epsilon^{(1)}\ll\cdots\ll \theta_\epsilon^{(q)}$ as $\epsilon\to0$, and the kernels $R_t^{(p)}:M_0\times M_0\to\mathbb{R}_+$, where $M_0$ denotes the set of local minima of $U$, such that \[ \lim_{\epsilon\to0}u_\epsilon(t\theta_\epsilon^{(p)},\,x)=\sum_{m'\in M_0}R_t^{(p)}(m,\,m')u_0(m'), \] for all $t>0$ and $x$ in the domain of attraction of $m$ for the dynamical system $\dot{x}(t)=b(x(t))$. We then complete the characterization of the solution $u_\epsilon$ by computing the exact asymptotic limit of the solution between time scales $\theta_\epsilon^{(p)}$ and $\theta_\epsilon^{(p+1)}$ for each $p$, where $\theta_\epsilon^{(0)}=1$ and $\theta_\epsilon^{(q+1)}=\infty$. Our analysis makes essential use of the hierarchical tree structure underlying the metastable behavior in different time-scales of the diffusion induced by $\mathscr{L}_\epsilon$. This result can be regarded as the precise refinement of Freidlin-Wentzell theory which was not known for more than a half century.

math.PR↗

arXiv2023

Metastability and time scales for parabolic equations with drift 1: the first time scale

Consider the elliptic operator given by $$ \mathscr{L}_{\epsilon}f= {b} \cdot \nabla f + \epsilon \Delta f $$ for some smooth vector field $ b\colon \mathbb R^d \to\mathbb R^d$ and a small parameter $\epsilon>0$. Consider the initial-valued problem $$ \left\{ \begin{aligned} &\partial_ t u_\epsilon = \mathscr L_\epsilon u_\epsilon,\\ &u_\epsilon (0, \cdot) = u_0(\cdot), \end{aligned} \right. $$ for some bounded continuous function $u_0$. Denote by $\mathcal M_0$ the set of critical points of $b$ which are stable stationary points for the ODE $\dot x (t) = b (x(t))$. Under the hypothesis that $\mathcal M_0$ is finite and $ b = -(\nabla U + \ell)$, where $ \ell$ is a divergence-free field orthogonal to $\nabla U$, the main result of this article states that there exist a time-scale $\theta^{(1)}_\epsilon$, $\theta^{(1)}_\epsilon \to \infty$ as $\epsilon \rightarrow 0$, and a Markov semigroup $\{p_t : t\ge 0\}$ defined on $\mathcal M_0$ such that $$ \lim_{\epsilon\to 0} u_\epsilon (t\theta^{(1)}_\epsilon, x) =\sum_{m'\in \mathcal M_0} p_t(m, m')\, u_0( m'), $$ for all $t>0$ and $ x$ in the domain of attraction of $m$ for the ODE $\dot{x}(t)= b( x(t))$. The time scale $\theta^{(1)}$ is critical in the sense that, for all time scale $\varrho_\epsilon$ such that $\varrho_\epsilon \to \infty$, $\varrho_\epsilon/\theta^{(1)}_\epsilon \to 0$, $$ \lim_{\epsilon\to 0} u_\epsilon (\varrho_\epsilon, x)=u_0(m) $$ for all $x \in \mathcal D(m)$. Namely, $\theta_\epsilon^{(1)}$ is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [Landim, Lee, Seo, forthcoming] we extend this result finding all critical time-scales at which the solution $u_\epsilon$ evolves smoothly in time and we show that the solution $u_\epsilon$ is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of $b$.

math.PR↗

arXiv2023

Dynamic and static large deviations of a one dimensional SSEP in weak contact with reservoirs

We derive a formula for the quasi-potential of one-dimensional symmetric exclusion process in weak contact with reservoirs. The interaction with the boundary is so weak that, in the diffusive scale, the density profile evolves as the one of the exclusion process with reflecting boundary conditions. In order to observe an evolution of the total mass, the process has to be observed in a longer time-scale, in which the density profile becomes immediately constant.

math.PR↗

arXiv2023

Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals

Let $\Xi_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets and consider a $\Xi_n$-valued, irreducible, reversible, continuous-time Markov chain $(X^{(n)}_t:t\ge 0)$. Denote by $\mathscr P(\mathbb R^d) $ the set of probability measures on $\mathbb R^d$ and by $I_n\colon \mathscr P(\mathbb R^d) \to [0,+\infty)$ the level two large deviations rate functional for $X^{(n)}_t$ as $t\to\infty$. We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of $I_n$ expressing it as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n)\, I^{(p)}$, where $I^{(p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$ represent rate functionals independent of $n$ and $\theta^{(p)}_n$ sequences such that $\theta^{(1)}_n \to\infty$, $\theta^{(p)}_n / \theta^{(p+1)}_n \to 0$ for $1\le p< q$. The speed $\theta^{(p)}_n$ corresponds to the time-scale at which the Markov chains $X^{(n)}_t$ exhibits a metastable behavior, and the $I^{(p-1)}$ zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.

math.PR↗

arXiv2022

Large deviations for diffusions: Donsker-Varadhan meet Freidlin-Wentzell

We consider a diffusion process on $\mathbb R^n$ and prove a large deviation principle for the empirical process in the joint limit in which the time window diverges and the noise vanishes. The corresponding rate function is given by the expectation of the Freidlin-Wentzell functional per unit of time. As an application of this result, we obtain a variational representation of the rate function for the Gallavotti-Cohen observable in the small noise and large time limits.

math.PR↗

arXiv2021

Thermodynamics of nonequilibrium driven diffusive systems in mild contact with boundary reservoirs

We consider macroscopic systems in weak contact with boundary reservoirs and under the action of external fields. We present an explicit formula for the Hamiltonian of such systems, from which we deduce the equation of motions, the action functional, the hydrodynamic equation for the adjoint dynamics, and a formula for the quasi-potential. We examine the case in which the external forcing depends on time and drives the system from one nonequilibrium state to another. We extend the results presented in [6] on thermodynamic transformations for systems in strong contact with boundary reservoirs to the present situation. In particular, we propose a natural definition of renormalized work, and show that it satisfies a Clausius inequality, and that quasi-static transformations minimize the renormalized work. In addition, we connect the renormalized work to the quasi-potential describing the fluctuations in the stationary nonequilibrium ensemble.

math.PR↗

arXiv2021

Concurrent Donsker-Varadhan and hydrodynamical large deviations

We consider the weakly asymmetric exclusion process on the $d$-dimensional torus. We prove a large deviations principle for the time averaged empirical density and current in the joint limit in which both the time interval and the number of particles diverge. This result is obtained both by analyzing the variational convergence, as the number of particles diverges, of the Donsker-Varadhan functional for the empirical process and by considering the large time behavior of the hydrodynamical rate function. The large deviations asymptotic of the time averaged current is then deduced by contraction principle. The structure of the minimizers of this variational problem corresponds to the possible occurrence of dynamical phase transitions.

math.PR↗