In the problem of the activation energy for a noise-induced transition over a finite given time in an arbitrary overdamped one-dimensional potential system, we find and classify all extremal paths and provide a simple algorithm to explicitly
select which is the most probable transition path (MPTP).
The activation energy is explicitly expressed in quadratures.
For the transition beyond the top of the barrier, the MPTP does not possess turning points and the activation energy is a monotonously decreasing function of the transition time. For transitions between points lying on one and the same slope of the potential well,
which may be relevant e.g. for the problem of the tails of the prehistory probability density, the situation is more complicated: the activation energy is a non-monotonous function of time and, most important, may possess bends corresponding to jump-wise switches in the topology of the MPTP; it can be proved also that the number of turning points in the MPTP is necessarily less than two. The prefactor is calculated numerically using the scheme suggested by Lehmann, Reimann and Hanggi, PRE 55, 419 (1998). The theory is compared with simulations.