Acta Applicandae Mathematicae

, Volume 145, Issue 1, pp 15–45 | Cite as

Quantitative Convergence Towards a Self-Similar Profile in an Age-Structured Renewal Equation for Subdiffusion

  • Hugues Berry
  • Thomas Lepoutre
  • Álvaro Mateos González


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 stationary 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 analytical solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent “pseudo-equilibrium”, which in turn converges to the stationary profile.


Age-structured PDE Renewal equation Anomalous diffusion Relative entropy estimates 

Mathematics Subject Classification

35Q92 92D25 60J75 35B40 


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Hugues Berry
    • 1
  • Thomas Lepoutre
    • 1
    • 2
  • Álvaro Mateos González
    • 1
    • 3
  1. 1.InriaVilleurbanneFrance
  2. 2.Institut Camille Jordan, CNRS UMR 5208Université de LyonVilleurbanne cedexFrance
  3. 3.Unité de Mathématiques Pures et Appliquées, CNRS UMR 5669Université de LyonLyonFrance

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