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Smoluchowski’s coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels. (English) Zbl 1072.60071

This paper is devoted to the study of Smoluchowski’s coagulation equation for constant, additive and multiplicative kernels. In the first section, the authors give some explanations of the heuristic motivations of the problem with a survey of results that already exist in the literature. In the second section, Smoluchowski’s coagulation equation with discrete mass is discussed. The authors give new proofs of the expressions of the solution in terms of discrete state branching processes. Their approach simplifies considerably the proofs in the literature given before. In the third section, the authors study Smoluchowski’s coagulation equation with continuous mass. They prove that the solution of the equation in the additive case and that for the multiplicative case are connected by some transformations given explicitly. Those results are used to establish the existence and uniqueness of the solutions. Some renormalization limit theorems for the solutions for constant, additive and multiplicative kernels are proved. The limit theorems show that after some scaling, a solution converges to a limit which depends on the initial condition only through its moments of order one, two and three.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
44A10 Laplace transform
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References:

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