Functional analysis
Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates
Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 753-759.

In the process of discrete and nonlocal aggregation, the major problem arises when each fragmentation rate becomes infinite at infinity. In this paper, a discrete Cauchy problem describing multiple fragmentation processes is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. We focus on the case where fragmentation rates are size and position dependent and where new particles are spatially randomly distributed according to a certain probabilistic law. Unlike [8], where the discrete model with bounded fragmentation rates is treated, we use, in this paper, Katoʼs theorem in L1 [2] and the dominated convergence theorem [4] to show the existence of the infinitesimal generator of a positive semigroup of contractions and give sufficient conditions for honesty in the case of unbounded fragmentation rates. Essentially, we demonstrate that, even in the discrete and nonlocal case, the process is conservative if at infinity daughter particles tend to go back into the system with a high probability.

Dans un processus dʼagrégation discret non local, un problème fondamental se pose lorsque chaque taux de fragmentation tend vers lʼinfini à lʼinfini. Dans cette Note, on étudie le problème de Cauchy discret dans le cas où les taux de fragmentation décrivent des processus de fragmentation multiple au moyen dʼopérateurs dépendant de paramètres et de la théorie des semi-groupes sous-stochastiques dépendant dʼun paramètre. On se concentre sur le cas où les taux de fragmentation dépendent de la dimension et de la position et où de nouvelles particules sont distribuées de manière aléatoire suivant une certaine loi de probabilité. À la différence de [8], qui traite dʼun modèle discret à taux de fragmentation borné, on utilise le théorème de Kato dans le cas L1 [2] et le théorème de la convergence dominée [4] pour démontrer lʼexistence dʼun générateur infinitésimal dʼun semi-groupe de contactions positif ; on donne des conditions suffisantes dʼhonnêteté dans le cas de taux de fragmentation non bornés. Fondamentalement, on démontre que, même dans le cas discret et non local, le processus est conservatif si, à lʼinfini, les particules filles tendent à rentrer dans le système avec une grande probabilité.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.09.023
Doungmo Goufo, Emile Franc 1; Oukouomi Noutchie, Suares Clovis 1

1 Department of Mathematical Sciences, North-West University, Mafikeng, South Africa
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Doungmo Goufo, Emile Franc; Oukouomi Noutchie, Suares Clovis. Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates. Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 753-759. doi : 10.1016/j.crma.2013.09.023. http://www.numdam.org/articles/10.1016/j.crma.2013.09.023/

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