  Behind the nice unification provided by the notion of the level 2.5 in the
field of large deviations for time-averages over a long Markov trajectory,
there are nevertheless very important qualitative differences between the
meaning of the level 2.5 for diffusion processes on one hand, and the meaning
of the level 2.5 for Markov chains either in discrete-time or in
continuous-time on the other hand. In order to analyze these differences in
detail, it is thus useful to consider two types of random walks converging
towards a given diffusion process in dimension $d$ involving arbitrary
space-dependent forces and diffusion coefficients, namely (i) continuous-time
random walks on the regular lattice of spacing $b$ ; (ii) discrete-time random
walks in continuous space with a small time-step $\tau$. One can then analyze
how the large deviations at level 2.5 for these two types of random walks
behave in the limits $b \to 0$ and $\tau \to 0$ respectively, in order to
describe how the fluctuations of some empirical observables of the random walks
are suppressed in the limit of diffusion processes. One can then also study the
limits $b \to 0$ and $\tau \to 0$ for any trajectory observable of the random
walks that can be decomposed on its empirical density and its empirical flows
in order to see how it is projected on the appropriate trajectory observable of
the diffusion process involving its empirical density and its empirical
current.
