On long time behavior of some coagulation processes. (Q2574631)

A random dynamical system of coalescing particles under the influence of Brownian excitations and subject to an attraction of a force proportional to the mass of a particle is considered. Particles move in the \(d\)-dimensional Euclidean space, and each of them consists of a finite number of elementary particles. The more elementary particles a single particle is constituted of, the smaller Brownian excitation it is subject to, and consequently, the more regular its motion is. Finally, two particles coalesce at a specified rate if they happen to occur at a fixed distance from each other. The problem is approached via spotting a position \(X_t\in \mathbb R^d\) and a mass \(M_t\in \mathbb N\) of a single particle in time \(t\), and studying the evolution of the process \((X_t,M_t)\). It is argued that the above dynamics of the spotted particle can be described by a system of stochastic differential equations (SDE) driven by a \(d\)-dimensional Wiener process and a Poisson measure with a certain intensity measure. The main results of the paper are the following: (1) The considered system of SDE has a solution for every suitably integrable initial condition if the coefficients of the SDE are sufficiently regular. (2) The mass \(M_t\) tends to infinity almost surely and the position \(X_t\) tends to zero in the mean-square, as \(t\) goes to infinity. (3) If additional restrictions upon the coefficient of the Brownian excitation and the coagulation kernel are imposed, the position process \(X_t\) is shown to tend to zero almost surely.