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

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 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.

Appendix

1.1 The Case $\mu=1$

It is quite interesting to notice that even if the behaviour is not really self-similar, our method gives a precise asymptotic for the case $\mu=1$. To illustrate this, we focus on the reference case: $\beta (a)=\frac{1}{1+a}$. In this case the ‘pseudo equilibrium reads’

$$W(\tau,b)=\frac{1}{(\mathrm{e}^{-\tau}+b)\log(1+\mathrm{e}^{\tau})}. $$

This pseudo equilibrium tends to a Dirac mass but gives still quantitative information. Indeed, following the same computation than for Eq. (17) for the case $\mu<1$, we obtain easily

$$\frac{\mathrm{d}}{ \mathrm{d}\tau} \int_{0}^{1} |w-W|\leq-\frac{\mathrm {e}^{\tau}}{1+\mathrm{e}^{\tau}} \int_{0}^{1} |w-W|+2 \bigl\vert C(\tau)\delta(\tau ) \bigr\vert . $$

Where we also have

$$C(\tau)\delta(\tau) = \int_{0}^{1} \biggl(\frac{\mathrm{e}^{\tau}b}{1+\mathrm {e}^{\tau}b}-1 \biggr)W( \tau,b)\mathrm{d}b. $$

This leads to

$$C(\tau)\delta(\tau)=- \int_{0}^{1} \frac{\mathrm{e}^{-\tau}}{(\mathrm {e}^{-\tau}+b)^{2}\log(1+\mathrm{e}^{\tau})}=\frac{\mathrm{e}^{-\tau}}{\log (1+\mathrm{e}^{\tau})} \biggl(\frac{1}{\mathrm{e}^{-\tau}+1}-\frac {1}{\mathrm{e}^{-\tau}} \biggr). $$

And finally,

$$C(\tau)\delta(\tau)=-\frac{\mathrm{e}^{\tau}}{(1+\mathrm{e}^{\tau})\log (1+\mathrm{e}^{\tau})}\rightarrow0. $$

And we can still claim that $\int_{0}^{1} |w-W|\rightarrow0$. We can give a (rough) estimate for a rate of convergence. Integrating, we have

$$\int_{0}^{1} |w-W|\leq\frac{1}{1+\mathrm{e}^{\tau}} \int_{0}^{1} |w-W|(\tau=0)+ 2 \frac{1}{1+\mathrm{e}^{\tau}} \int_{0}^{\tau}\frac{\mathrm{e}^{\tau'}}{\log (1+\mathrm{e}^{\tau'})}{\mathrm{d}} \tau' . $$

We estimate the second term

$$\frac{1}{1+\mathrm{e}^{\tau}} \int_{0}^{\tau}\frac{\mathrm{e}^{\tau'}}{\log (1+\mathrm{e}^{\tau'})}{\mathrm{d}} \tau'= \frac{1}{1+\mathrm{e}^{\tau}} \int_{1}^{\mathrm{e}^{\tau}} \frac{1}{\log(1+u)}{\mathrm{d}}u. $$

This term behaves as $1 / \tau$. Indeed, we have easily (splitting the integral at $\mathrm{e}^{\alpha\tau}$ for $\alpha<1$):

$$\begin{aligned} \frac{1}{\log(1+\mathrm{e}^{\tau})} \leq&\frac{1}{1+\mathrm{e}^{\tau}} \int _{1}^{\mathrm{e}^{\tau}} \frac{1}{\log(1+u)}{\mathrm{d}}u\leq \frac{\mathrm {e}^{(\alpha-1) \tau}}{1+\mathrm{e}^{-\tau}}+\frac{1}{\log(1+\mathrm {e}^{\alpha\tau})} \\ \int_{0}^{1} |w-W| \leq&\frac{1}{1+\mathrm{e}^{\tau}} \int_{0}^{1} |w-W|(\tau=0)+ \frac{K}{1+\tau}\leq \frac{K'}{1+\tau} . \end{aligned}$$