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

• Published in: Acta applicandae mathematicae Vol. 145; no. 1; pp. 15 - 45
• Main Authors: Berry, Hugues, Lepoutre, Thomas, González, Álvaro Mateos
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-10-2016; Springer Nature B.V
• Subjects: Age; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Cells (biology); Computational Mathematics and Numerical Analysis; Convergence; Evolution; Exact solutions; Mathematical analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Probability Theory and Stochastic Processes; Random walk; Self-similarity; Time dependence; 60J75; 92D25; Age-structured PDE; Anomalous diffusion; 35B40; Renewal equation; Relative entropy estimates; 35Q92
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-016-0048-3

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.