  The longstanding question of how stochastic behaviour arises from
deterministic Hamiltonian dynamics is of great importance, and any truly
holistic theory must be capable of describing this transition. In this review,
we introduce the influence functional formalism in both the quantum and
classical regimes. Using this technique, we demonstrate how irreversible
behaviour arises generically from the reduced microscopic dynamics of a
system-environment amalgam. The influence functional is then used to rigorously
derive stochastic equations of motion from a microscopic Hamiltonian. In this
method stochastic terms are not identified heuristically, but instead arise
from an exact mapping only available in the path-integral formalism. The
interpretability of the individual stochastic trajectories arising from the
mapping is also discussed. As a consequence of these results, we are also able
to show that the proper classical limit of stochastic quantum dynamics
corresponds non-trivially to a generalised Langevin equation derived with the
classical influence functional. This provides a further unifying link between
open quantum systems and their classical equivalent, highlighting the utility
of influence functionals and their potential as a tool in both fundamental and
applied research.
