Glauco Valle
We obtain the hydrodynamic limit of one-dimensional interacting particle systems describing the macroscopic evolution of the density of mass in infinite volume from the microscopic dynamics. The processes are weak pertubations of the symmetric exclusion process by shift operators describing the spread of particles around positions where new sites are created.
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We prove a uniform Poincare' inequality for non-interacting unbounded spin systems with a conservation law, when the single-site potential is a bounded perturbation of a convex function. The result is then applied to Ginzburg-Landau processes to show diffusive scaling of the associated spectral gap.
We derive the hydrodynamic limit of the Kawasaki dynamics for the one-dimensional conservative system of unbounded real-valued spins with arbitrary strong, quadratic and finite-range interactions. This extends prior results for non-interacting spin systems. The result is obtained by adapting two scale approach of Grunewald, Otto, Villani and Westdickenberg combined with the authors' recent approach on conservative systems with strong interactions.
Hydrodynamic behavior near dynamical criticality of a facilitated conservative lattice gas
本文研究二维保守格子气体的动力学活跃-吸收相变，推导其流体力学方程，证明临界指数可由两个独立指数确定，并发现超临界趋近临界点时出现与亚临界不同的尺度分离现象。
Conservative stochastic PDEs on the whole space
本文为全空间上的保守随机偏微分方程建立适定性理论，涵盖带相关噪声的Dean--Kawasaki方程，并严格化零程粒子过程中涨落流体力学与宏观涨落理论的联系。
A central limit theorem for nonlinear conservative SPDEs
本文证明了适用于非线性退化扩散及非正则噪声系数的保守型随机偏微分方程的中心极限定理，并揭示其涨落与相互作用粒子系统流体力学极限涨落的一阶一致性。
We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on ${\mathbb Z}^n$, where the jump rates are asymmetric and long-range of order $\|x\|^{-(n+α)}$ for a particle displacement of order $\|x\|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<α<1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $α\geq 1$, we derive a Burgers hydrodynamic equation, as in the fini…
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.
Degree-preserving conservative processes and a unified approach for their hydrodynamics
本文研究一类满足“保度性质”的一维守恒型相互作用系统，建立了不变测度与流体动力学极限的统一分析框架，证明乘积不变测度的边缘分布属于六种特定分布族，并揭示宏观行为由经典热方程控制。
We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is attached to a point on the left and there is a force (tension) $τ$ acting on the right. In order to provide good ergodic properties to the system, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy, converge to the solution of the Euler system of equations, in the smooth regime.
We study the hydrodynamic limits of the simple exclusion processes and the zero range processes on crystal lattices. For a periodic realization of crystal lattice, we derive the hydrodynamic limit for the exclusion processes and the zero range processes, which depends on both the structure of crystal lattice and the periodic realization. Even through the crystal lattices have inhomogeneous local structure, for all periodic realizations, we apply the entropy method to derive the hydrodynamic limits. Also, we discuss how the limit equation depends on the choices of the realizations.
Motivated by the recent preprint [arXiv:2004.08412] by Ayala, Carinci, and Redig, we first provide a general framework for the study of scaling limits of higher order fields. Then, by considering the same class of infinite interacting particle systems as in [arXiv:2004.08412], namely symmetric simple exclusion and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [arXiv:2004.0…
We analyze certain conservative interacting particle system and establish ergodicity of the system for a family of invariant measures. Furthermore, we show that convergence rate to equilibrium is exponential. This result is of interest because it presents counterexample to the standard assumption of physicists that conservative system implies polynomial rate of convergence.
Hydrodynamic limit of the directed exclusion process
研究定向排斥过程的水力学极限，通过显式多过程耦合证明其宏观稳定性与吸引力，探讨其开放性问题。
Tractable description of hydrodynamic limits of a class of interacting jump processes on sparse graphs
本文针对稀疏随机图上的马尔可夫相互作用跳跃过程，在单粒子转移无环假设下，用有限耦合常微分方程组给出邻域经验测度流体动力学极限的可处理描述，并通过模拟验证其近似效果。
We consider an interacting unbounded spin system, with conservation of the mean spin. We derive quantitative rates of convergence to the hydrodynamic limit provided the single-site potential is a bounded perturbation of a strictly convex function with polynomial growth, and with an additional random inhomogeneous linear term. This additional linear term models the impact of a random chemical potential. The argument adapts the two-scale approach of Grunewald, Otto, Villani and Westdickenberg from the quadratic to the general case. The main ingredient is the derivation of a covariance estimate t…
We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution of a regularized version of PDE.
We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erdős-Rényi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs ("decorated" with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of …
Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived Markovian coupled process $(ξ_t,ζ_t)_{t\geq 0}$ satisfies: (A) if $ξ_0\leqζ_0$ (coordinate-wise), then for all $t\geq 0$, $ξ_t\leqζ_t$ a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on $\Z^d$ such that, in each transition, $k$ particles may jump from a site $x$ to another site $y$, with $k\geq 1$. These models include simple exclusion for wh…
We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an original interacting particle system for which we discuss the propagation of chaos. We consider a time-discretized approximation of this particle system to which we associate a random function which is proved to converge to a solution of a regularized version of a nonlinear PDE.
This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential 1/N . This mimics the effect of chemical syna…
Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process
本文提出带非线性保守噪声的 SPDE 逼近序列，刻画对称简单排斥过程在流体动力学极限附近的涨落与稀有事件，并证明零噪声大偏差速率函数一致。
We consider attractive irreducible conservative particle systems on $\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1--30], from which we relax the assumption that the process has explicit invariant measures.
Mean field limits of a class of conservative systems with position-dependent transition rates
本文研究一类含非对称排他过程和零程过程的保守系统，证明其流体动力学极限由非线性常微分方程驱动，并在粒子数有界时证明涨落由广义Ornstein-Uhlenbeck过程驱动，关键步骤是用图形方法证明不同位置粒子数近似独立。
This is a review based on the presentation done at the seminar Laurent Schwartz in December 2021. It is announcing results in the forthcoming [Menegaki-Mouhot-Marahrens'22]. This work presents a new simple quantitative method for proving the hydrodynamic limit of a class of interacting particle systems on lattices. We present here this method in a simplified setting, for the zero-range process and the Ginzburg-Landau process with Kawasaki dynamics, in the parabolic scaling and in dimension $1$. The rate of convergence is quantitative and uniform in time. The proof relies on a consistence-stabi…
We consider systems of mean-field interacting diffusions, where the pairwise interaction structure is described by a sparse (and potentially inhomogeneous) random graph. Examples include the stochastic Kuramoto model with pairwise interactions given by an Erdős-Rényi graph. Our problem is to compare the bulk behavior of such systems with that of corresponding systems with dense nonrandom interactions. For a broad class of interaction functions, we find the optimal sparsity condition that implies that the two systems have the same hydrodynamic limit, which is given by a McKean-Vlasov diffusion.…
This is the first of two articles on the study of a particle system model that exhibits a Turing instability type effect. The model is based on two discrete lines (or toruses) with Ising spins, that evolve according to a continuous time Markov process defined in terms of macroscopic Kac potentials and local interactions. For fixed time, we prove that the density fields weakly converge to the solution of a system of partial differential equations involving convolutions. The presence of local interactions results in the lack of propagation of chaos, reason why the hydrodynamic limit cannot be ob…
Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions
本文为广义Dean--Kawasaki随机偏微分方程建立中心极限定理与大偏差原理，刻画其小噪声波动，并推广至一般$C^2$有界域上的Dirichlet边界条件，对应粒子边界吸收或注入。
This paper is a follow-up of the work initiated in [3], where it has been investigated the hydrodynamic limit of symmetric independent random walkers with birth at the origin and death at the rightmost occupied site. Here we obtain two further results: first we characterize the stationary states on the hydrodynamic time scale and show that they are given by a family of linear macroscopic profiles whose parameters are determined by the current reservoirs and the system mass. Then we prove the existence of a super-hyrdrodynamic time scale, beyond the hydrodynamic one. On this larger time scale t…
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.
A new non-conservative stochastic reaction-diffusion system in which two families of random walks in two adjacent domains interact near the interface is introduced and studied in this paper. Such a system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. We show that in the macroscopic limit, the particle densities converge to the solution of a coupled nonlinear heat equations. For this, we first prove that propagation of chaos holds by establishing the uniqueness of a new BBGKY hierarchy.…
Hydrodynamic limit for a chain with thermal and mechanical boundary forces
本文研究一维开放谐振子链在边界恒温器与外部张力作用下的流体动力学极限，证明体积拉伸和能量经验剖面在扩散尺度下收敛于非线性扩散保守偏微分方程组的解。
Stochastic ordering, attractiveness and couplings in non-conservative particle systems
本文研究了允许个体多次到达、离开和非保守跳跃的相互作用粒子系统的随机比较，获得了随机比较与吸引性的充分必要条件，证明了其与保序 Markovian 耦合的等价性，并将耦合表示为网络流问题的解。
This paper concerns with the hydrodynamic limit of the Kob-Andersen model, an interacting particle system that has been introduced by physicists in order to explain glassy behavior, and widely studies since. We will see that the density profile evolves in the hydrodynamic limit according to a non-degenerate hydrodynamic equation, and see how the diffusion coefficient decays as density grows.
The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions of SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves th…
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