  Ultraslow diffusion (i.e. logarithmic diffusion) has been extensively studied
theoretically, but has hardly been observed empirically. In this paper,
firstly, we find the ultraslow-like diffusion of the time-series of word counts
of already popular words by analysing three different nationwide language
databases: (i) newspaper articles (Japanese), (ii) blog articles (Japanese),
and (iii) page views of Wikipedia (English, French, Chinese, and Japanese).
Secondly, we use theoretical analysis to show that this diffusion is basically
explained by the random walk model with the power-law forgetting with the
exponent $\beta \approx 0.5$, which is related to the fractional Langevin
equation. The exponent $\beta$ characterises the speed of forgetting and $\beta
\approx 0.5$ corresponds to (i) the border (or thresholds) between the
stationary and the nonstationary and (ii) the right-in-the-middle dynamics
between the IID noise for $\beta=1$ and the normal random walk for $\beta=0$.
Thirdly, the generative model of the time-series of word counts of already
popular words, which is a kind of Poisson process with the Poisson parameter
sampled by the above-mentioned random walk model, can almost reproduce not only
the empirical mean-squared displacement but also the power spectrum density and
the probability density function.
