  We suggest that the broad distribution of time scales in financial markets
could be a crucial ingredient to reproduce realistic price dynamics in stylised
Agent-Based Models. We propose a fractional reaction-diffusion model for the
dynamics of latent liquidity in financial markets, where agents are very
heterogeneous in terms of their characteristic frequencies. Several features of
our model are amenable to an exact analytical treatment. We find in particular
that the impact is a concave function of the transacted volume (aka the
"square-root impact law"), as in the normal diffusion limit. However, the
impact kernel decays as $t^{-\beta}$ with $\beta=1/2$ in the diffusive case,
which is inconsistent with market efficiency. In the sub-diffusive case the
decay exponent $\beta$ takes any value in $[0,1/2]$, and can be tuned to match
the empirical value $\beta \approx 1/4$. Numerical simulations confirm our
theoretical results. Several extensions of the model are suggested.
