Alexander David Gottlieb
We establish a necessary and sufficient condition for the propagation of chaos by a family of many-particle Markov processes, if the particles live in a Polish space: a sequence of n-particle Markov transition functions propagates chaos if and only if it propagates chaos for pure initial states.
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We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions.…
We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly.
We establish the Strong Poisson Hypothesis for symmetric closed networks. In particular, the asymptotic independence of the nodes -- as the size of the system tends to infinity -- is proved.
By Girsanov's thoerem and using the existing log-Harnack inequality for distribution independent SDEs, the log-Harnack inequality is derived for path-distribution dependent stochastic Hamiltonian systems. As an application, the exponential ergodicity in relative entropy is obtained by combining with transportation cost inequality. In addition, the quantitative propagation of chaos in the sense of Wasserstein distance, which together with the coupling by change of measure implies the quantitative propagation of chaos in total variation norm as well as relative entropy are obtained.
Propagation of Chaos for Nonlinear Markov Chains
本文研究非线性马尔可夫链的传播混沌，结合非线性马尔可夫理论，提出定量经验测度估计和传播混沌分析，并应用于标准McKean-Vlasov扩散和粒子过滤。
High-Temperature Increasing Propagation of Chaos and its breakdown for the Hopfield Model
该研究探讨了Hopfield模型在高温区域中传播混沌的增加及其破裂现象，揭示了不同参数条件下混沌行为的变化规律。
The trend to equilibrium in large time is studied for a large particle system associated to a Vlasov-Fokker-Planck equation in the presence of a convex external potential, without smallness restriction on the interaction. From this are derived uniform in time propagation of chaos estimates, which themselves yield in turn an exponentially fast convergence for the semi-linear equation itself. The approach is quantitative.
In this paper, we first investigate the well-posedness of a backward stochastic differential equation where the driver depends on the law of the solution conditioned to a common noise. Under standard assumptions, we show that existence and uniqueness, as well as integrability results, still hold. We also study the associated interacting particles system, for which we prove propagation of chaos, with quantitative estimates on the rate of convergence in Wasserstein distance.
The propagation of chaos is a central concept of kinetic theory that serves to relate the equations of Boltzmann and Vlasov to the dynamics of many-particle systems. Propagation of chaos means that molecular chaos, i.e., the stochastic independence of two random particles in a many-particle system, persists in time, as the number of particles tends to infinity. We establish a necessary and sufficient condition for a family of general n-particle Markov processes to propagate chaos. This condition is expressed in terms of the Markov transition functions associated to the n-particle processes, an…
In this paper, we develop a numerical approach based on Chaos expansions to analyze the sensitivity and the propagation of epistemic uncertainty through a queueing systems with breakdowns. Here, the quantity of interest is the stationary distribution of the model, which is a function of uncertain parameters. Polynomial chaos provide an efficient alternative to more traditional Monte Carlo simulations for modelling the propagation of uncertainty arising from those parameters. Furthermore, Polynomial chaos expansion affords a natural framework for computing Sobol' indices. Such indices give reli…
Propagation of Chaos for Weakly Interacting Mild Solutions to Stochastic Partial Differential Equations
本文研究系数不满足Lipschitz条件的半线性随机偏微分方程弱相互作用温和解的混沌传播性质，并给出McKean-Vlasov随机偏微分方程温和解存在唯一性的连续性与线性增长条件。
This paper focus on investigating the explicit rate of convergence for the propagation of chaos, in a pathwise sense a family of interacting stochastic particle related to some Brownian driven McKean-Vlasov dynamics. Precisely the McKean form of nonlinearity is concentrated on a path dependent drift component and satisfies a particular sub-gaussian moment control. Such control enables to derive a uniform estimate of the cost in terms of exponential martingale between the particle and its McKean/mean-field limit system which in turn provide an optimal rate of propagation of chaos in terms of th…
Epstein introduced an agent-based model called Demographic Prisoner's dilemma. He shows, via simulations, that cooperation in this spatial evolutionary repeated game can be sustained. In order to do proves, we put on this model a particle system framework in order to prove the convergence of the spatial model to a random matching model, using homogenization techniques. Then we prove the convergence of the random matching model to a mean field model, using propagation of chaos techniques.
Entropy-cost type Propagation of Chaos for Mean Field Particle System with Bounded Measurable Interaction
本文研究了均场相互作用粒子系统中熵-成本类型混沌传播，仅假设相互作用有界可测，且单粒子初始分布在由某种成本函数Ψ诱导的L^1运输成本W_1^Ψ下收敛于对应的麦肯-瓦拉索夫SDEs分布。
Consider $N$ balls initially placed in $L$ bins. At each time step take a ball from each non-empty bin and \emph{randomly} reassign the balls into the bins.We call this finite Markov chain \emph{General Repeated Balls into Bins} process. It is a discrete time interacting particles system with parallel updates. Assuming a \emph{quantitative} chaotic condition on the reassignment rule we prove a \emph{quantitative} propagation of chaos for this model. We furthermore study some equilibrium properties of the limiting nonlinear process.
Long Time Entropy-Cost type Propagation of Chaos
本文利用随机噪声的正则化效应，提出了平均场粒子系统的定量熵-成本型混沌传播，证明任意正时刻的相对熵混沌性质仅依赖于较弱的初始 $L^2$-Wasserstein 距离，并在耗散假设下及路径相关情形下均成立。
Sharp propagation of chaos in Rényi divergence
本文研究了相互作用扩散系统在平稳状态下的Rényi散度中混沌传播的精确速率，通过熵层次和等周性条件，得到了q-Rényi散度的界。
In this note, we clarify the well-posedness of the limit equations to the mean-field $N$-neuron models proposed in Baladron et al. and we prove the associated propagation of chaos property. We also complete the modeling issue in Baladron et al. by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters.
Quantitative propagation of chaos for particle systems with memory and their long-time behaviour
本文用耦合方法研究具有记忆的相互作用扩散粒子系统，建立了定量混沌传播估计，并证明在适当条件下大粒子极限与长时间极限可交换。
Well-posedness and propagation of chaos for jump-type McKean-Vlasov SDEs with irregular coefficients
本文研究跳跃型McKean-Vlasov SDE在不规则系数下的强解适定性，并利用Yamada-Watanabe逼近和停时获得Euler格式的混沌传播与收敛速度。
Sharp Propagation of Chaos for the Ensemble Langevin Sampler
本文研究ensemble Langevin采样器的混沌传播，采用同步耦合与基于停时的技术处理经验协方差，证明了具有最优速率的路径意义混沌传播。
We prove the existence of a contraction rate for Vlasov-Fokker-Planck equation in Wasserstein distance, provided the interaction potential is (locally) Lipschitz continuous and the confining potential is both Lipschitz continuous and greater than a quadratic function, thus requiring no convexity conditions. Our strategy relies on coupling methods suggested by A. Eberle adapted to the kinetic setting enabling also to obtain uniform in time propagation of chaos in a non convex setting.
Conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching: wellposedness, propagation of chaos, averaging principle
本文研究带跳和马尔可夫机制切换的条件McKean-Vlasov随机微分方程，利用L2-Wasserstein距离证明强适定性，建立平均场粒子系统的混沌传播及显式收敛速率，并针对双时间尺度情形建立平均化原理。
In this paper we obtain uniform propagation estimates for systems of interacting diffusions. We adopt a general model, satisfying various conditions which ensure that the decay resulting from the internal dynamics term dominates the interaction and noise terms. The result should have diverse applications, particularly in neuroscience, and allows for models more elaborate than that of Wilson and Cowan, not requiring the internal dynamics to be of linear decay.
We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the ($k$-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in rough path spaces and allows for a robust subsequent analysis of the particle system and its McKean-Vlasov type limit, as shown in two corollaries.
We prove existence and uniqueness of a reaction-diffusion equation whose diffusivity is a non-linear functional of the boundary temperature. We do this by studying systems of one-dimensional reflecting diffusions whose noise is a function of the reflection local-time of the system, and by characterizing the large-scale (hydrodynamic) behavior by showing propagation of chaos. In addition, we analyze the one-particle case by computing the distribution of the hitting times of its reflection local-time. This work is the noise analog of work done by Frank Knight (2001).
Propagation of chaos for maxima of particle systems with mean-field drift interaction
研究大规模粒子系统中归一化极值的渐进行为，证明在大群体极限下，极值行为类似于独立同分布系统中每个粒子遵循麦肯-瓦拉索夫极限动力学的情况。
In 2001, Knight constructed a stochastic process modeling the one dimensional interaction of two particles, one being Newtonian in the sense that it obeys Newton's laws of motion, and the other particle being Brownian. We construct a multi-particle analog, using Skorohod map estimates in proving a propagation of chaos and characterizing the hydrodynamic limit as the solution to a PDE with free boundary condition. Stochastic methods are used to show existence and uniqueness for the free boundary problem, and also present an algorithm of approximating the solution.
Consider a finite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This finite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable (chaotic) set of initial states, as $L\to+\infty$, the numbers of balls in each bin becomes independent from the rest of the system i.e. we have propagation of chaos. We furthermore study some equilibrium properties of the limiting nonline…
Volterra type McKean-Vlasov SDEs with singular kernels: Well-posedness, Propagation of Chaos and Euler schemes
本文研究奇异核Volterra型McKean-Vlasov SDE，建立了适定性，证明了带显式收敛率的混沌传播，并提出了粒子系统的显式Euler格式。
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.
Conditional McKean-Vlasov Differential Equations with Common Poissonian Noise: Propagation of Chaos
本文研究受公共泊松噪声和布朗运动驱动的大群体相互作用系统，建立其平均场McKean-Vlasov极限，并在群体规模趋于无穷时证明了解的弱收敛性，从而获得混沌传播结果。
In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincaré inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.
The Bird and Nanbu systems are particle systems used to approximate the solution of the mollied Boltzmann equation. In particular, they have the propagation of chaos property. Following [GM94, GM97, GM99], we use coupling techniques and results on branching processes to write an expansion of the error in the propagation of chaos in terms of the number of particles, for slightly more general systems than the ones cited above. This result leads to the proof of the a.s convergence and the centrallimit theorem for these systems. In particular, we have a central-limit theorem for the empirical meas…
Uniform in Time Propagation of Chaos for Mean Field Particle System with Interacting Noise and Partially Dissipative Drifts
本文针对具有相互作用噪声和部分耗散漂移的平均场粒子系统，利用反射耦合、梯度估计和Duhamel公式，在$L^1$-Wasserstein距离下建立了时间上一致的定量混沌传播。
Distribution Dependent Birth-Death Processes: $\mathbb{W}_p$-Estimate, Ergodicity and Propagation of Chaos
本文研究了时间非齐次分布依赖出生-死亡过程的well-posedness、$\mathbb{W}_p$-估计、指数遍历性及时间一致的混沌传播，扩展了分布依赖SDE和均场粒子系统的相关结果。
Mean-field branching SDEs: propagation of chaos, scaling limits and phase transitions
该研究针对带依赖律迁入的分支SDE及其平均场粒子近似，建立混沌传播界、两阶段尺度极限，给出逻辑斯蒂平均场扩散灭绝/非灭绝判据，明确弱相互作用是时间一致近似的必要条件。
Spatially structured neural networks driven by jump diffusion noise with monotone coefficients, fully path dependent delay and with a disorder parameter are considered. Well-posedness for the associated McKean-Vlasov equation and a corresponding propagation of chaos result in the infinite population limit are proven. Our existence result for the McKean-Vlasov equation is based on the Euler approximation, that is applied to this type of equation for the first time.
Stability of backward propagation of chaos
本文提出混沌反向传播相对于初始数据集的稳定性定义，并通过数据集收敛与函数连续性框架，证明离散到连续鞅驱动的平均场及 McKean-Vlasov BSDE 的稳定收敛。
Propagation of Chaos for Derivatives of McKean-Vlasov Stochastic Differential Equations and Applications
本文研究了麦金-沃拉茨夫随机微分方程导数的混沌传播，证明了初始值扰动和驱动噪声扰动下的收敛性，并给出了可精确的收敛速率。
Sharp propagation of chaos for McKean-Vlasov equation with non constant diffusion coefficient
本文利用 BBGKY 层级和相对熵方法，将既有混沌传播结果推广到扩散系数非常数且可依赖经验测度的 N 粒子系统。
Quantitative Propagation of Chaos in $L^η(η\in(0,1])$-Wasserstein distance for Mean Field Interacting Particle System
本文推导了平均场相互作用粒子系统在$L^η$-Wasserstein距离下的混沌定量传播，允许扩散系数相互作用且初始分布收敛，通过解耦SDEs的梯度估计分别研究了非退化与二阶系统。
Propagation of chaos for point processes induced by particle systems with mean-field drift interaction
研究由具有均场漂移相互作用的粒子系统诱导的点过程的渐进行为，证明在合适条件下该点过程的混沌传播，其弱极限与独立同分布的麦肯维拉索方程解的点过程相同，弱极限为强度测度与经典极值分布相关的泊松点过程，特别地，得到归一化上序统计量的极限分布。
This note shows how to considerably strengthen the usual mode of convergence of an $n$-particle system to its McKean-Vlasov limit, often known as propagation of chaos, when the volatility coefficient is nondegenerate and involves no interaction term. Notably, the empirical measure converges in a much stronger topology than weak convergence, and any fixed $k$ particles converge in total variation to their limit law as $n\rightarrow\infty$. This requires minimal continuity for the drift in both the space and measure variables. The proofs are purely probabilistic and rather short, relying on Girs…
Propagation of chaos in Fisher information
研究具有光滑相互作用和漂移粒子在费希尔信息中的混沌传播，利用新引理及精细估计，通过BBGKY层次结构得到微分不等式系统，实现混沌传播推广，且通过高斯例子证明衰减率最优。
Propagation of chaos for stochastic particle systems with singular mean-field interaction of $L^q-L^p$ type
本文利用 Partial Girsanov 变换，在相互作用核属于 $L_t^q-L_x^p$ 空间的条件下，证明了平均场随机粒子系统的适定性与混沌传播。
This article is concerned with the exponential stability and the uniform propagation of chaos properties of a class of Extended Ensemble Kalman-Bucy filters with respect to the time horizon. This class of nonlinear filters can be interpreted as the conditional expectations of nonlinear McKean Vlasov type diffusions with respect to the observation process. In contrast with more conventional Langevin nonlinear drift type processes, the mean field interaction is encapsulated in the covariance matrix of the diffusion. The main results discussed in the article are quantitative estimates of the expo…
Propagation of chaos is a well-studied phenomenon and shows that weakly interacting diffusions may become independent as the system size converges to infinity. Most of the literature focuses on the case of exchangeable systems where all involved diffusions have the same distribution and are "of the same size". In this paper, we analyze the case where only a few diffusions start outside of an accessible trap. Our main result shows that in this "sparse regime" the system of weakly interacting diffusions converges in distribution to a forest of excursions from the trap. In particular, initial ind…
Propagation of chaos in the Random field Curie-Weiss model
本文证明了随机场 Curie-Weiss 模型在顺磁相及相变点处的混沌传播，并分析了自由能多极小值情形下系统随机选择平衡测度时混沌传播仍成立。
Chaos propagation in genetic algorithms: An optimal transport approach
本文将遗传算法视为粒子系统，证明其在大量粒子极限下由玻尔兹曼型方程描述，采用最优传输方法证明混沌传播结果，并以Wasserstein-1距离最优速率收敛。
We prove a quantitative propagation of chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Guérin and Méléard \cite{FonGueMe} and Fournier \cite{Fournier} where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.
Well-posedness and propagation of chaos for L{é}vy-driven McKean-Vlasov SDEs under Lipschitz assumptions
本文证明Lévy驱动的McKean-Vlasov SDE在Lipschitz条件下的强适定性，并给出路径级定量混沌传播结果，同时改进线性相互作用及α-稳定噪声情形的收敛速率。
We study a stochastic spatial epidemic model where the $N$ individuals carry two features: a position and an infection state, interact and move in $\R^d$. In this Markovian model, the evolution of the infection states are described with the help of the Poisson Point Processes , whereas the displacement of the individuals are driven by mean field advection, a (state dependence) diffusion and also a common noise, so that the spatial dynamic is a random process. We prove that when the number $N$ of individual goes to infinity, the conditional propagation of chaos holds : conditionnally to the com…
Long Time $\W_0$-$\widetilde{\W}_1$ type Propagation of Chaos for Mean Field Interacting Particle System
本文研究Lévy噪声驱动的平均场相互作用粒子系统的长时间混沌传播问题，通过耦合方法推导$\text{W}_0$-$\tilde{\text{W}}_1$型结果，并将其应用于乘性布朗运动和加性α-稳定噪声驱动的系统。
Propagation of chaos for moderately interacting particle systems related to singular kinetic McKean-Vlasov SDEs
本文研究中等相互作用粒子系统中混沌传播性质，用于近似由alpha稳定过程驱动的奇异动能麦肯-瓦拉斯斯德过程。
Quantitative weak propagation of chaos for McKean--Vlasov branching diffusion processes
本文研究了具有分支的 McKean-Vlasov 扩散过程的弱传播混沌，通过 Itô 公式和函数导数估计，推导了有限种群分支扩散过程的定量化弱收敛速率。
Emergence of Common Noise: Quantitative Conditional Propagation of Chaos
研究具有平均场相互作用和通过集体随机扰动产生的共同噪声的粒子系统，利用随机流和随机缝补引理建立离散Euler格式及其连续时间流极限的混沌条件量化传播估计，得到显式Wasserstein收敛率。
We consider the 3D Landau equation for moderately soft potentials ($γ\in(-2,0)$ with the usual notation) as well as a stochastic system of $N$ particles approximating it. We first establish some strong/weak stability estimates for the Landau equation, which are satisfying only when $γ\in [-1,0)$. We next prove, under some appropriate conditions on the initial data, the so-called propagation of molecular chaos, i.e. that the empirical measure of the particle system converges to the unique solution of the Landau equation. The main difficulty is the presence of a singularity in the equation. When…
Metastability and Sharp Propagation-of-Chaos Thresholds for Mean-Field Interacting Diffusions on the Circle
本文研究圆上平均场相互作用扩散的亚稳定性和混沌传播，确定了尖锐的指数时间阈值，并给出了临界轨道有限性的可检验准则，应用于Kuramoto模型和Transformer交互。
Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics
本文在平均场能量泛函凸性假设下，利用次 coercivity 方法证明动力学平均场 Langevin 动力学的指数收敛及均匀时间混沌传播，并应用于两层神经网络训练。
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