In this talk, I will discuss an exact mapping between many-body quantum spin systems and classical stochastic processes. This approach can handle integrable and non-integrable systems, including those in higher dimensions, in a unified framework, and can be applied both in and out of equilibrium. Focussing on quantum quenches, I will discuss dynamical quantum phase transitions in the Loschmidt amplitude, showing that these correspond to enhanced fluctuations and other features in the classical stochastic coordinates