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We study the hitting times of Markov processes to target set $G$, startingfrom a reference configuration $x_0$ or its basin of attraction. Theconfiguration $x_0$ can correspond to the bottom of a (meta)stable well, whilethe target $G$ could be either a set of saddle (exit) points of the well, or aset of further (meta)stable configurations. Three types of results arereported: (1) A general theory is developed, based on the path-wise approach tometastability, which has three important attributes. First, it is general inthat it does not assume reversibility of the process, does not focus only onhitting 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 andexplicit bounds on the corrections to exponentiality. (2) We compare and relatedifferent metastability conditions proposed in the literature so to eliminatepotential sources of confusion. This is specially relevant for evolutions ofinfinite-volume systems, whose treatment depends on whether and how relevantparameters (temperature, fields) are adjusted. (3) We introduce the notion ofearly asymptotic exponential behavior to control time scales asymptoticallysmaller than the mean-time scale. This control is particularly relevant forsystems with unbounded state space where nucleations leading to exit frommetastability can happen anywhere in the volume. We provide natural sufficientconditions on recurrence times for this early exponentiality to hold and showthat it leads to estimations of probability density functions.

Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility