The dynamics of an infinite continuum system of randomly jumping and coalescing point particles is studied. The states of the system are probability measures on the corresponding configuration space Γ the evolution of which is constructed in the following way. The evolution of observables F0 → Ft is obtained from a Kolmogorovtype evolution equation. Then the evolution of states µ0 → µt is defined by the relation µ0(Ft) = µt(F0) for F0 belonging to a measure-defining class of functions. The main result of the paper is the proof of the existence of the evolution of this type for a bounded time horizon.2010 Mathematics Subject Classification. 60K35; 82C22.

A continuum individual-based model of hopping and coalescing particles is introduced and studied. Its microscopic dynamics are described by a hierarchy of evolution equations obtained in the paper. Then the passage from the micro-to mesoscopic dynamics is performed by means of a Vlasov-type scaling. The existence and uniqueness of the solutions of the corresponding kinetic equation are proved.2010 Mathematics Subject Classification. 60K35; 35Q83; 82C22.

No abstract

An algorithm is proposed for finding numerical solutions of a kinetic equation that describes an infinite system of point particles placed in $\mathbb {R}^{d} (d \geq 1)$ ℝ d ( d ≥ 1 ) . The particles perform random jumps with pair-wise repulsion in the course of which they can also merge. The kinetic equation is an essentially nonlinear and nonlocal integro-differential equation, which can hardly be solved analytically. The numerical algorithm which we use to solve it is based on a space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, and adjustable system-size schemes. We show that, for special choices of the model parameters, the solutions manifest unusual time behavior. A numerical error analysis of the obtained results is also carried out.

No abstract