  Following on from our recent work, we investigate a stochastic approach to
non-equilibrium quantum spin systems. We show how the method can be applied to
a variety of physical observables and for different initial conditions. We
provide exact formulae of broad applicability for the time-dependence of
expectation values and correlation functions following a quantum quench in
terms of averages over classical stochastic processes. We further explore the
behavior of the classical stochastic variables in the presence of dynamical
quantum phase transitions, including results for their distributions and
correlation functions. We provide details on the numerical solution of the
associated stochastic differential equations, and examine the growth of
fluctuations in the classical description. We discuss the strengths and
limitations of the current implementation of the stochastic approach and the
potential for further development.
