Quantitative convergence towards a self similar profile in an age-structured renewal equation for subdiffusion

Hugues Berry 1, 2, 3 Thomas Lepoutre 4, 5 Álvaro Mateos González 6, 7, 1
1 BEAGLE - Artificial Evolution and Computational Biology
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information, Inria Grenoble - Rhône-Alpes, LBBE - Laboratoire de Biométrie et Biologie Evolutive, CarMeN - Cardiovasculaire, métabolisme, diabétologie et nutrition
4 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
ICJ - Institut Camille Jordan [Villeurbanne], Inria Grenoble - Rhône-Alpes, CGPhiMC - Centre de génétique et de physiologie moléculaire et cellulaire
7 NUMED - Numerical Medicine
Inria Grenoble - Rhône-Alpes, UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
Abstract : Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the subdiffusive continuous-time random walk using age-structured partial differential equations with age renewal upon each walker jump, where the age of a walker is the time elapsed since its last jump. In the spatially-homogeneous (zero-dimensional) case, we follow the evolution in time of the age distribution. An approach inspired by relative entropy techniques allows us to obtain quantitative explicit rates for the convergence of the age distribution to a self-similar profile, which corresponds to convergence to a stationnary profile for the rescaled variables. An important difficulty arises from the fact that the equation in self-similar variables is not autonomous and we do not have a specific analyitcal solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent "pseudo-equilibrium", which in turn converges to the stationnary profile.
Type de document :
Article dans une revue
Acta Applicandae Mathematicae, Springer Verlag, 2016, pp.15-45
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https://hal.inria.fr/hal-01136667
Contributeur : Álvaro Mateos González <>
Soumis le : vendredi 27 mars 2015 - 17:41:06
Dernière modification le : mardi 17 juillet 2018 - 15:50:24
Document(s) archivé(s) le : mardi 18 avril 2017 - 02:22:23

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  • HAL Id : hal-01136667, version 1
  • ARXIV : 1503.08552

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Hugues Berry, Thomas Lepoutre, Álvaro Mateos González. Quantitative convergence towards a self similar profile in an age-structured renewal equation for subdiffusion. Acta Applicandae Mathematicae, Springer Verlag, 2016, pp.15-45. 〈hal-01136667〉

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