  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.
