  This work is devoted to show an equivalent description for the most probable
transition paths of stochastic dynamical systems with Brownian noise, based on
the theory of Markovian bridges. The most probable transition path for a
stochastic dynamical system is the minimizer of the Onsager-Machlup action
functional, and thus determined by the Euler-Lagrange equation (a second order
differential equation with initial-terminal conditions) via a variational
principle. After showing that the Onsager-Machlup action functional can be
derived from a Markovian bridge process, we first demonstrate that, in some
special cases, the most probable transition paths can be determined by first
order deterministic differential equations with only a initial condition. Then
we show that for general nonlinear stochastic systems with small noise, the
most probable transition paths can be well approximated by solving a first
order differential equation or an integro differential equation on a certain
time interval. Finally, we illustrate our results with several examples.
