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

Hugues Berry 1, 2, 3 Thomas Lepoutre 4, 5, 6 Álvaro Mateos González 7, 8, 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 - UMR 5558
4 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
CGPhiMC - Centre de génétique et de physiologie moléculaire et cellulaire, Inria Grenoble - Rhône-Alpes, ICJ - Institut Camille Jordan [Villeurbanne]
6 MMCS - Modélisation mathématique, calcul scientifique
ICJ - Institut Camille Jordan [Villeurbanne]
8 NUMED - Numerical Medicine
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées, Inria Grenoble - Rhône-Alpes
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.
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https://hal.inria.fr/hal-01136667
Contributor : Álvaro Mateos González <>
Submitted on : Friday, March 27, 2015 - 5:41:06 PM
Last modification on : Friday, August 28, 2020 - 11:16:09 PM
Long-term archiving on: : Tuesday, April 18, 2017 - 2:22:23 AM

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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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