  A Langevin equation whose deterministic part undergoes a saddle-node
bifurcation is investigated theoretically. It is found that statistical
properties of relaxation trajectories in this system exhibit divergent
behaviors near a saddle-node bifurcation point in the weak-noise limit, while
the final value of the deterministic solution changes discontinuously at the
point. A systematic formulation for analyzing a path probability measure is
constructed on the basis of a singular perturbation method. In this
formulation, the critical nature turns out to originate from the neutrality of
exiting time from a saddle-point. The theoretical calculation explains results
of numerical simulations.
