Traveling waves and homogeneous fragmentation
We formulate the notion of the classical Fisher-Kolmogorov-Petrovskii- Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of class...
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| Format: | Journal article |
| Language: | English |
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2011
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| author | Berestycki, J Harris, S Kyprianou, A |
| author_facet | Berestycki, J Harris, S Kyprianou, A |
| author_sort | Berestycki, J |
| collection | OXFORD |
| description | We formulate the notion of the classical Fisher-Kolmogorov-Petrovskii- Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323-331] and [Comm. Pure Appl. Math. 29 (1976) 553-554], Neveu [In Seminar on Stochastic Processes (1988) 223-242 Birkhäuser] and Chauvin [Ann. Probab. 19 (1991) 1195-1205], our analysis exposes the relation between traveling waves and certain additive and multiplicative martingales via laws of large numbers which have been previously studied in the context of Crump-Mode-Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57 (1981) 365-395] and in the context of fragmentation processes by Bertoin and Martinez [Adv. in Appl. Probab. 37 (2005) 553-570] and Harris, Knobloch and Kyprianou [Ann. Inst. H. Poincaré Probab. Statist. 46 (2010) 119-134]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion (cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503-517] and Biggins and Kyprianou [Electr. J. Probab. 10 (2005) 609-631]) showing their mathematical robustness even within the context of fragmentation theory. © 2011 Institute of Mathematical Statistics. |
| first_indexed | 2024-03-07T01:19:28Z |
| format | Journal article |
| id | oxford-uuid:8fd8fa83-15f9-43cb-b89b-f82329f9c4fa |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:19:28Z |
| publishDate | 2011 |
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| spelling | oxford-uuid:8fd8fa83-15f9-43cb-b89b-f82329f9c4fa2022-03-26T23:07:16ZTraveling waves and homogeneous fragmentationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:8fd8fa83-15f9-43cb-b89b-f82329f9c4faEnglishSymplectic Elements at Oxford2011Berestycki, JHarris, SKyprianou, AWe formulate the notion of the classical Fisher-Kolmogorov-Petrovskii- Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323-331] and [Comm. Pure Appl. Math. 29 (1976) 553-554], Neveu [In Seminar on Stochastic Processes (1988) 223-242 Birkhäuser] and Chauvin [Ann. Probab. 19 (1991) 1195-1205], our analysis exposes the relation between traveling waves and certain additive and multiplicative martingales via laws of large numbers which have been previously studied in the context of Crump-Mode-Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57 (1981) 365-395] and in the context of fragmentation processes by Bertoin and Martinez [Adv. in Appl. Probab. 37 (2005) 553-570] and Harris, Knobloch and Kyprianou [Ann. Inst. H. Poincaré Probab. Statist. 46 (2010) 119-134]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion (cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503-517] and Biggins and Kyprianou [Electr. J. Probab. 10 (2005) 609-631]) showing their mathematical robustness even within the context of fragmentation theory. © 2011 Institute of Mathematical Statistics. |
| spellingShingle | Berestycki, J Harris, S Kyprianou, A Traveling waves and homogeneous fragmentation |
| title | Traveling waves and homogeneous fragmentation |
| title_full | Traveling waves and homogeneous fragmentation |
| title_fullStr | Traveling waves and homogeneous fragmentation |
| title_full_unstemmed | Traveling waves and homogeneous fragmentation |
| title_short | Traveling waves and homogeneous fragmentation |
| title_sort | traveling waves and homogeneous fragmentation |
| work_keys_str_mv | AT berestyckij travelingwavesandhomogeneousfragmentation AT harriss travelingwavesandhomogeneousfragmentation AT kyprianoua travelingwavesandhomogeneousfragmentation |