Large deviations conditioned on large deviations I: Markov chain and Langevin equation

We present a systematic analysis of stochastic processes conditioned on an empirical observable Q(T) defined in a time interval [0, T], for large T. We build our analysis starting with a discrete time Markov chain. Results for a continuous time Markov process and Langevin dynamics are derived as limiting cases. In the large T limit, we show how conditioning on a value of Q(T) modifies the dynamics. For a Langevin dynamics with weak noise and conditioned on Q(T), we introduce large deviation functions and calculate them using either a WKB method or a variational formulation. This allows us, in particular, to calculate the typical trajectory and the fluctuations around this trajectory when conditioned on a certain value of Q(T), for large T.