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Stochastic semigroups and coagulation equations. (English) Zbl 1093.60070

Ukr. Mat. Zh. 57, No. 6, 770-777 (2005) and Ukr. Math. J. 57, No. 6, 913-922 (2005).
The author considers equations of the form \[ \partial_tf=Q[f](r),\quad t>0,r>0, \] where \[ Q[f](r)=\frac{1}2 \int_0^\infty \int_0^\infty A(r;r_1,r_2)\alpha (r_1,r_2)f(r_1)f(r_2)\,dr_1\,dr_2- f(r)\int_0^\infty \alpha (r,r_1)f(r_1)\,dr_1, \] describing coagulation of clusters in various physical situations. Here \(f\) is the density of clusters of size \(r\), \(\alpha (r,r_1)\) is the coagulation rate, \(A\) is the weighted probability that the interaction of a cluster of size \(r_1\) and another cluster of size \(r_2\) generates a cluster of size \(r\). It is shown that solutions of such equations can be approximated by solutions of stochastic systems describing the coagulation process in terms of Markov semigroups describing dynamics of a system of interacting particles; see M. Lachowicz and M. Pulvirenti [Arch. Ration. Mech. Anal. 109, No. 1, 81–93 (1990; Zbl 0682.76002)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Citations:

Zbl 0682.76002
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