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 with 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.