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Suppose that particles are randomly distributed in $\bR^d$, and they aresubject to identical stochastic motion independently of each other. TheSmoluchowski process describes fluctuations of the number of particles in anobservation region over time. This paper studies properties of the Smoluchowskiprocesses and considers related statistical problems. In the first part of thepaper we revisit probabilistic properties of the Smoluchowski process in aunified and principled way: explicit formulas for generating functionals andmoments are derived, conditions for stationarity and Gaussian approximation arediscussed, and relations to other stochastic models are highlighted. The secondpart deals with statistics of the Smoluchowki processes. We consider twodifferent models of the particle displacement process: the undeviated uniformmotion (when a particle moves with random constant velocity along a straightline) and the Brownian motion displacement. In the setting of the undeviateduniform motion we study the problems of estimating the mean speed and the speeddistribution, while for the Brownian displacement model the problem ofestimating the diffusion coefficient is considered. In all these settings wedevelop estimators with provable accuracy guarantees.

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