Classical statistical mechanics usually approximates the classical N-body dynamics by dissipative stochastic equations for macroscopic variables, which are either of the form of master equations (generalizing Markov chains to an infinite number of states) or of the form of stochastic differential equations. In both cases we end up with Markov processes. Their stationary states lead to equilibrium thermodynamics.
Quantum statistical mechanics usually approximates the quantum N-body or quantum field dynamics by dissipative stochastic equations for measurable variables which are only sometimes classical Markov processes. In general they have rather the form of quantum stochastic processes, which are generalizations of classical Markov processes to the case of noncommuting quantities.
However, more refined models lead in both cases to non-Markovian classical or quantum stochastic dynamics with memory. Usually the memory last for a very short time only and hence can be neglected. But for some materials the memory is long-living enough to be of industrial value.
Useful references are books by Accardi et al. (Quantum theory and its stochastic limit, 2013), Breuer & Petruccione (The theory of open quantum systems, 2002). or Gardiner and Zoller (Quantum noise, 2004).