Suppose that particles are randomly distributed in $\bR^d$, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. This paper studies properties of the Smoluchowski processes and considers related statistical problems. In the first part of the paper we revisit probabilistic properties of the Smoluchowski process in a unified and principled way: explicit formulas for generating functionals and moments are derived, conditions for stationarity and Gaussian approximation are discussed, and relations to other stochastic models are highlighted. The second part deals with statistics of the Smoluchowki processes. We consider two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In the setting of the undeviated uniform motion we study the problems of estimating the mean speed and the speed distribution, while for the Brownian displacement model the problem of estimating the diffusion coefficient is considered. In all these settings we develop estimators with provable accuracy guarantees.
翻译:假设粒子随机地以$\bR ⁇ d$进行分布, 并且它们彼此独立地受到相同的随机运动的影响。 Smoluchowski 过程描述一个观察区域的粒子数量随时间变化的波动情况。 本文研究Smolchowski 过程的特性, 并考虑相关的统计问题。 在论文的第一部分, 我们以统一和有原则的方式重新审视Smolchowski 过程的概率性能: 生成功能和时间的清晰公式被推导出来, 讨论固定性和高斯近似的条件, 并突出与其他随机模型的关系。 第二部分涉及Smolchowki 过程的统计。 我们考虑两种不同的粒子迁移过程模式: 不可避免的统一运动( 当粒子随直线随机恒定速度移动时) 和布朗运动的迁移。 在确定未经证实的统一运动时, 我们研究估算平均速度和速度分布的问题, 而对于布朗移位模型来说, 估计扩散系数的问题被考虑。 在所有这些环境中, 我们用精确度来进行估计。