  We present an extended version of the recently proposed "LLOB" model for the
dynamics of latent liquidity in financial markets. By allowing for finite
cancellation and deposition rates within a continuous reaction-diffusion setup,
we account for finite memory effects on the dynamics of the latent order book.
We compute in particular the finite memory corrections to the square root
impact law, as well as the impact decay and the permanent impact of a
meta-order. The latter is found to be linear in the traded volume and
independent of the trading rate, as dictated by no-arbitrage arguments. In
addition, we consider the case of a spectrum of cancellation and deposition
rates, which allows us to obtain a square root impact law for moderate
participation rates, as observed empirically. Our multi-scale framework also
provides an alternative solution to the so-called price diffusivity puzzle in
the presence of a long-range correlated order flow.
