沿向量场的循环有限速度随机运动的扩散极限
Diffusion limits of cyclic finite-velocity random motions along vector fields
- Friedrich Schiller University Jena（耶拿弗里德里希·席勒大学）
- Institute of Mathematics, Friedrich Schiller University Jena（耶拿弗里德里希·席勒大学数学研究所）
- Institute of Geophysics, National Academy of Sciences of Ukraine（乌克兰国家科学院地球物理研究所）
- Igor Sikorsky Kyiv Polytechnic Institute（伊戈尔·西科尔斯基基辅理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对受主动粒子模型启发的循环有限速度随机运动，在Kac型标度下证明其弱收敛到多维扩散，揭示循环顺序、动力学类型对扩散极限的影响及几何因素的作用。
AI中文摘要：
我们研究受主动粒子模型启发的一类多元非均匀有限速度随机运动的扩散近似。粒子在规定的速度场$V_1,\dots,V_p$之间循环交替，具有随机运行时间，且根据两种动力学演化：要么是由分段线性运动构成的“飞行”动力学，要么是粒子遵循对应速度流的“滑动”动力学。在将消失的运行时间与发散的粒子速度耦合的Kac型标度下，我们证明了两种过程都弱收敛到多维扩散过程。结果表明，连续循环运动的顺序会影响扩散极限。除了由随机运行时间波动产生的二阶项外，极限漂移还包含基础向量场的方向导数$DV_i[V_j]$。在Stratonovich形式中，该漂移的一部分通过它们的李括号$[V_i,V_j]$表示。因此，微观运动的一般非交换性在扩散标度下得以保留，并产生依赖于循环顺序的宏观漂移。尽管飞行和滑动动力学由相同的向量场和运行时间驱动，但极限漂移也对二者进行了区分。若干示例，包括“跑-退”运动和由多个向量场产生的循环动力学，说明了循环切换协议和基础速度场的几何形状如何塑造极限扩散。
英文摘要：
We investigate diffusion approximations for a class of multivariate inhomogeneous finite-velocity random motions motivated by models of active particles. A particle alternates cyclically among prescribed velocity fields $V_1,\dots,V_p$ with random run times and evolves according to either a \emph{flight} dynamics, consisting of piecewise-linear motion, or a \emph{gliding} dynamics, in which the particle follows the corresponding velocity flow. Under a Kac-type scaling that couples vanishing run times with diverging particle speed, we prove weak convergence of both processes to multidimensional diffusions. It turns out that the order of successive cyclic motions affects the diffusion limit. In addition to the second-order term generated by fluctuations of the random run times, the limiting drift contains directional derivatives $DV_i[V_j]$ of the underlying vector fields. In Stratonovich form, part of this drift is expressed through their Lie brackets $[V_i,V_j]$. Thus, the generic non-commutativity of the microscopic motions survives the diffusive scaling and generates a macroscopic drift which depends on the cyclic order. The limiting drift also distinguishes the flight and gliding dynamics, despite their being driven by the same vector fields and run times. Several examples, including run-and-reverse motion and cyclic dynamics generated by multiple vector fields, illustrate how the cyclic switching protocol and the geometry of the underlying velocity fields shape the limiting diffusions.
