具有镜面边界条件和环境噪声的群聚模型
Swarming models with specular boundary condition and environmental noise
AI总结：
本文研究带反射边界与共同环境噪声的随机相互作用粒子群聚模型，通过紧性论证严格推导新型随机偏微分平均场方程。
摘要：
我们研究了在有界区域内用于群聚/自集体行为的一类通用模型。该模型表示为一个随机相互作用粒子系统，同时受到反射边界条件和共同环境噪声的影响。我们严格推导了其相应的宏观平均场方程；由于共同噪声的存在，这是一种新型的随机偏微分方程。该方法依赖于一个紧性论证：我们首先建立与粒子系统相关的经验测度的紧性，随后证明极限测度的时间边缘分布是该平均场方程的一个解。
英文摘要：
We investigate a general class of models for swarming/self-collective behaviour in domains with boundaries. The model is expressed as a stochastic system of interacting particles subject to both reflecting boundary condition and common environmental noise. We rigorously derive its corresponding macroscopic mean-field equation, which is a new type of stochastic partial differential equation due to the presence of common noise. The approach relies on a compactness argument, in which we first establish the tightness of the empirical measures associated with the particle system and then demonstrate that the time marginal of the limit measure is a solution to the mean-field equation.
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