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August  2005, 5(3): 529-542. doi: 10.3934/dcdsb.2005.5.529

Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes

Jacek Banasiak 1, and  Mustapha Mokhtar-Kharroubi 2,

1. 

School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041
2. 

Laboratoire de Mathématiques, Université de Franche-Comté, Route de Gray, 25030 Besançon Cedex, France

Received  April 2004 Revised  June 2004 Published  May 2005

By a dishonest process we understand a process in which, for some initial data, there occurs an unaccounted for loss of the described quan- tity throughout the evolution. Classical examples are offered by shattering fragmentation, where the total mass is decreasing faster than predicted by the formal conservation laws, or explosive birth-and-death processes which, being formally conservative, suffer from the loss of individuals in the course of evo- lution. In this note we shall show, for these two processes, that if dishonesty occurs for one initial datum, then it must occur for any of them.
Keywords: shattering, substochastic semigroups, birth-and-death processes., fragmentation.
Mathematics Subject Classification: 47D06, 34G10, 45K05, 35F10, 92D25, 60J8.
Citation: Jacek Banasiak, Mustapha Mokhtar-Kharroubi. Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 529-542. doi: 10.3934/dcdsb.2005.5.529
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