  The most probable transition paths of a stochastic dynamical system are the
global minimizers of the Onsager-Machlup action functional and can be described
by a necessary but not sufficient condition, the Euler-Lagrange equation (a
second-order differential equation with initial-terminal conditions) from a
variational principle. This work is devoted to showing a sufficient and
necessary characterization for the most probable transition paths of stochastic
dynamical systems with Brownian noise. We prove that, under appropriate
conditions, the most probable transition paths are completely determined by a
first-order ordinary differential equation. The equivalence is established by
showing that the Onsager-Machlup action functional of the original system can
be derived from the corresponding Markovian bridge process. For linear
stochastic systems and the nonlinear Hongler's model, the first-order
differential equations determining the most probable transition paths are shown
analytically to imply the Euler-Lagrange equations of the Onsager-Machlup
functional. For general nonlinear systems, the determining first-order
differential equations can be approximated, in a short time or for the small
noise case. Some numerical experiments are presented to illustrate our results.
