Smoluchowski processes and nonparametric estimation of functionals of particle displacement distributions from count data

TL;DR

This paper analyzes the probabilistic properties of Smoluchowski processes and introduces nonparametric estimators for particle displacement functionals, including mean speed, speed distribution, and diffusion coefficient, with proven accuracy guarantees.

Contribution

It provides a unified probabilistic framework for Smoluchowski processes and develops novel nonparametric estimators for key particle displacement parameters.

Findings

Explicit formulas for generating functionals and moments of Smoluchowski processes

Conditions for stationarity and Gaussian approximation established

Consistent estimators with accuracy guarantees for mean speed, speed distribution, and diffusion coefficient

Abstract

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.