  We study the hitting times of Markov processes to target set $G$, starting
from a reference configuration $x_0$ or its basin of attraction. The
configuration $x_0$ can correspond to the bottom of a (meta)stable well, while
the target $G$ could be either a set of saddle (exit) points of the well, or a
set of further (meta)stable configurations. Three types of results are
reported: (1) A general theory is developed, based on the path-wise approach to
metastability, which has three important attributes. First, it is general in
that it does not assume reversibility of the process, does not focus only on
hitting times to rare events and does not assume a particular starting measure.
Second, it relies only on the natural hypothesis that the mean hitting time to
$G$ is asymptotically longer than the mean recurrence time to $x_0$ or $G$.
Third, despite its mathematical simplicity, the approach yields precise and
explicit bounds on the corrections to exponentiality. (2) We compare and relate
different metastability conditions proposed in the literature so to eliminate
potential sources of confusion. This is specially relevant for evolutions of
infinite-volume systems, whose treatment depends on whether and how relevant
parameters (temperature, fields) are adjusted. (3) We introduce the notion of
early asymptotic exponential behavior to control time scales asymptotically
smaller than the mean-time scale. This control is particularly relevant for
systems with unbounded state space where nucleations leading to exit from
metastability can happen anywhere in the volume. We provide natural sufficient
conditions on recurrence times for this early exponentiality to hold and show
that it leads to estimations of probability density functions.
