Weak and strong mean-field limits for stochastic Cucker-Smale particle systems

等级领导与随机互动下的Cucker-Smale群集行为

Cucker-Smale Flocking Under Hierarchical Leadership and Random Interactions

考虑一群鸟在它们之间相互作用。的相互作用是通过一个分层系统中，每只鸟，在每个时间步，调整自己的速度，根据他过去的速度和加权平均值的相对速度的上级在层次结构中。我们考虑另外一个事实，即每只鸟都有一定的概率看不到它的上级，这取决于它们之间的距离。对于这个模型的随机相互作用，我们证明了群集现象，获得类似的确定性模型，持有真实的。

On Stochastic Cucker-Smale flocking dynamics

We present a stochastic version of the Cucker-Smale flocking dynamics based on a markovian $N$-particle system of pair interactions with unbounded and, in general, non-Lipschitz continuous interaction potential. We establish the infinite particle limit $N \to \infty$ and identify the limit as a solution with a nonlinear martingale problem describing the law of a weak solution to a Vlasov-McKean stochastic equation with jumps. Moreover, we estimate the total variation and Wasserstein distance for the time-marginals from which uniqueness in the class of solutions having some finite exponential m…

Flocking and concentration behavior for the stochastic Cucker-Smale system in a harmonic field

We consider the Cucker-Smale system with multiplicative noise in a harmonic potential field and investigate the effect of harmonic potential field. In the presence of external potential force, the system is expected to emerge into almost surely velocity flocking and spatial concentration, due to the alignment mechanism, confining harmonic potential field and multiplicative noise. By constructing a stochastic Lyapunov functional, we derive a sufficient condition for the almost surely flocking and concentration behavior for the stochastic particle model, and verify it numerically. Then, we discu…

Flocking of the Cucker-Smale and Motsch-Tadmor models on general weighted digraphs via a probabilistic method

In this paper, we discuss the flocking phenomenon for the Cucker-Smale and Motsch-Tadmor models in continuous time on a general oriented and weighted graph with a general communication function. We present a new approach for studying this problem based on a probabilistic interpretation of the solutions. We provide flocking results under four assumptions on the interaction matrix and we highlight how they relate to the convergence in total variation of a certain Markov jump process. Indeed, we refine previous results on the minimal case where the graph admits a unique closed communication class…

基于随机批处理方法的Cucker-Smale模型的平均场极限

On the mean-field limit of the Cucker-Smale model with Random Batch Method

本文研究Cucker-Smale模型的随机批处理方法平均场极限，分别处理粒子数和离散时间间隔极限，用Wasserstein距离量化差异，并提出保持正性与动量的RBM-gPC方法近似随机平均场方程。

关于 Cucker-Smale 趋同动力学的降阶惯性 PDE 模型

On reduced inertial PDE models for Cucker-Smale flocking dynamics

本文提出一种仅依赖粒子位置的两方程降阶惯性PDE模型，直接在经验密度层面进行降阶以连接底层粒子动力学，证明其满足趋同连续定义并量化了与粒子系统的差异。

在嘈杂的环境中群集

Flocking in noisy environments

我们考虑一个扰动版本的动态介绍了Cucker和Smale（“涌现行为的羊群”），并证明，在类似的条件下，近对齐（一个概念，是精确的文本）是实现了一定的概率，从下面有界。

重尾Lévy型噪声扰动下Morse-Smale动力系统的亚稳定性

Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed Lévy type noise

我们考虑一类有限维确定性动力系统，它具有许多局部吸引子$K^i$，每个吸引子支持一个唯一的遍历概率测度$P^i$，其中特别包括任意有限维的Morse-Smale系统.动力学系统被小振幅$\varepsilon>0$的乘性非高斯重尾Lévy型噪声扰动。具体来说，我们考虑扰动导致伊藤，Stratonovich和规范（马库斯）随机微分方程。在$\varepsilon \to 0$的极限中向内指向的向量场的情况下，来自吸引力$D^i$的每个域的相应的渐近第一退出时间和位置问题由作者在[J. Stoch.一个. 32（1），163-190，2014]。我们将这些结果扩展到具有特征边界的域，并表明扰动系统在存在唯一的$\varepsilon$依赖时间尺度的意义上表现出亚稳态行为，在该时间尺度上，随机系统收敛到在不变测度$P^i$之间切换的连续时间马尔可夫链。作为例子，我们考虑$α$ -稳定的扰动的Duffing方程和化学系统表现出的birrhythmic行为。

随机平均场极限：非Lipschitz力\& Swarming

Stochastic Mean-Field Limit: Non-Lipschitz Forces \& Swarming

我们考虑与噪声相互作用的一般粒子随机系统，这些系统与动物集体行为的模型相关，并严格证明在平均场极限下，该系统接近动力学偏微分方程的解。我们的目标是包括在文献中广泛研究的模型，如Cucker-Smale模型，为个体行为添加噪声。的困难，相比经典的情况下，全球Lipschitz潜力，是在几个模型中的粒子之间的相互作用潜力只是局部Lipschitz，当地Lipschitz常数增长到无穷大的区域的大小考虑。考虑到这一点，我们提出了一个扩展的经典理论的全局Lipschitz相互作用，它只适用于局部Lipschitz的。

Effects of water currents on fish migration through a Feynman-type path integral approach under $\sqrt{8/3}$ Liouville-like quantum gravity surfaces

A stochastic differential game theoretic model has been proposed to determine optimal behavior of a fish while migrating against water currents both in rivers and oceans. Then, a dynamic objective function is maximized subject to two stochastic dynamics, one represents its location and another its relative velocity against water currents. In relative velocity stochastic dynamics, a Cucker-Smale type stochastic differential equation is introduced under white noise. As the information regarding hydrodynamic environment is incomplete and imperfect, a Feynman type path integral under $\sqrt{8/3}$ …

确定性和随机 flocking 模型的均场极限：非线性速度对齐

Mean-Field Limits of Deterministic and Stochastic Flocking Models with Nonlinear Velocity Alignment

本文研究了具有非线性速度对齐的 flocking 模型的均场极限，扩展了 Cucker-Smale 理论到非线性框架，并在确定性和随机性情况下提供了收敛率的改进。

Emergence of oscillatory behaviors for excitable systems with noise and mean-field interaction, a slow-fast dynamics approach

We consider the long-time dynamics of a general class of nonlinear Fokker-Planck equations, describing the large population behavior of mean-field interacting units. The main motivation of this work concerns the case where the individual dynamics is excitable, i.e. when each isolated dynamics rests in a stable state, whereas a sufficiently strong perturbation induces a large excursion in the phase space. We address the question of the emergence of oscillatory behaviors induced by noise and interaction in such systems. We tackle this problem by considering this model as a slow-fast system (the …

交互粒子系统中高效的在线学习

Efficient Online Learning in Interacting Particle Systems

本文提出了一种在交互粒子系统中进行高效在线学习的方法，通过连续观测少量粒子并递归更新模型参数，实现了对渐近对数似然的收敛性证明，并通过数值实验验证了其有效性。

On Limit Measures and Their Supports for Stochastic Ordinary Differential Equations

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin-Wentzell or Dembo-Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This…