Relevance of initial and final conditions for the Fluctuation Relation in Markov processes

Numerical observations on a Markov chain and on the continuous Markov process performed by a granular tracer show that the ``usual'' fluctuation relation for a given observable is not verified for finite (but arbitrarily large) times. This suggests that some terms which are usually expected to be negligible, i.e. ``border terms'' dependent only on initial and final states, in fact cannot be neglected. Furthermore, the Markov chain and the granular tracer behave in a quite similar fashion.