  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.
