I was reading Why the Principle of Least Action? and the top voted answer says
You can go further mathematically by learning the path integral formulation of nonrelativistic quantum mechanics and seeing how it leads to high probability for paths of stationary action. [emphasis added]
That seems to imply that, while paths of stationary action are most probable, it's theoretically possible for some other path to be taken. That reminds me of the second law of thermodynamics and how it's fundamentally a probabilistic law - it's not actually guaranteed that the total entropy of a closed system will approach a maximum in any finite amount of time, it's just that the probability of this happening quickly approaches 100% as the size of the system and/or the amount of time increases. Is something similar the case with the principle of stationary action in general?
That is, could we, at least in principle, explain all instances of the principle of stationary action holding (in classical mechanics, QM, and QFT) in terms of probability? Or is this something specific to the path integral formulation of QM and the probability interpretation wouldn't make sense in other models where we apply the principle?
