I will present a statistical theory of reactive trajectories between two pre-specified sets A and B, i.e. the portions of the path of a
Markov process during which the path makes a transition from A to B. This
problem is relevant e.g. in the context of metastability, in which case
the two sets A and B are metastable sets, though the formalism we propose is independent of any such assumptions on A and B. Various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so-called committor functions which give the probability that the process reaches first B before reaching A, either backward or forward in time. Using these objects, the following can be obtained: (i) the distribution of reactive trajectories, which gives the proportion of time reactive trajectories spend in sets outside of A and B; (ii) the hitting point distribution of the reactive trajectories the integral of which on a surface gives the net average flux
of reactive trajectories across this surface; and (iv) the average frequency of reactive trajectories, which gives the average number of transitions between A and B per unit of time. Time permitting, I will also illustrated how this theoretical framework can be used to design a numerical algorithm, termed the string method, to identify the committor function in high
dimensional systems.