  This work is devoted to the investigation of the most probable transition
time between metastable states for stochastic dynamical systems. Such a system
is modeled by a stochastic differential equation with non-vanishing Brownian
noise, and is restricted in a domain with absorbing boundary. Instead of
minimizing the Onsager-Machlup action functional, we examine the maximum
probability that the solution process of the system stays in a neighborhood (or
a tube) of a transition path, in order to characterize the most probable
transition path. We first establish the exponential decay lower bound and a
power law decay upper bound for the maximum of this probability. Based on these
estimates, we further derive the lower and upper bounds for the most probable
transition time, under suitable conditions. Finally, we illustrate our results
in simple stochastic dynamical systems, and highlight the relation with some
relevant works.
