  We study the trajectories of a solution $X_t$ to an It\^o stochastic
differential equation in $\Rm^d$, as the process passes between two disjoint
open sets, $A$ and $B$. These segments of the trajectory are called transition
paths or reactive trajectories, and they are of interest in the study of
chemical reactions and thermally activated processes. In that context, the sets
$A$ and $B$ represent reactant and product states. Our main results describe
the probability law of these transition paths in terms of a transition path
process $Y_t$, which is a strong solution to an auxiliary SDE having a singular
drift term. We also show that statistics of the transition path process may be
recovered by empirical sampling of the original process $X_t$. As an
application of these ideas, we prove various representation formulas for
statistics of the transition paths. We also identify the density and current of
transition paths. Our results fit into the framework of the transition path
theory by E and Vanden-Eijnden.
