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The inference of Markov models from data on stochastic dynamical trajectoriesover the large time-window $T$ is revisited via the Large Deviations at Level2.5 for the time-empirical density and the time-empirical flows. The goal is toobtain the large deviations properties for the probability distribution of theinferred Markov parameters in order to characterize their possible fluctuationsaround the true Markov parameters for large $T$. The explicit rate functionsare given for several settings, namely discrete-time Markov chains,continuous-time Markov jump processes, and diffusion processes in dimension$d$. Applications to various models of random walks in disordered media aredescribed, where the goal is to infer the quenched disordered variablesdefining a given disordered sample.

Inference of Markov models from trajectories via Large Deviations at Level 2.5 with applications to random walks in disordered media