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

Large deviations at level 2.5 and for trajectories observables of  diffusion processes : the missing parts with respect to their random-walks  counterparts