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

TL;DR

This paper studies the convergence of age-structured models for subdiffusive processes in cells, providing explicit rates for how the age distribution approaches a self-similar profile using entropy methods.

Contribution

It introduces a quantitative approach to analyze convergence towards self-similarity in age-structured subdiffusion models, overcoming the challenge of non-autonomous equations.

Findings

Explicit convergence rates to self-similar profiles

Quantitative estimates of attraction to pseudo-equilibria

Application of entropy techniques to age-structured PDEs

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.