  We generalize the classical theory of Brownian motion so as to reckon with
non-Markovian effects on both Klein-Kramers and Smoluchowski equations. For a
free particle and a harmonic oscillator, it is shown that such non-Markovian
effects account for the differentiability of the Brownian trajectories as well
as the breakdown of the energy equipartition of statistical mechanics at short
times in some physical situations. This non-Markovian approach is also extended
to look at anomalous diffusion. Next, we bring in the dynamical-quantization
method for investigating open quantum systems, which does consist in quantizing
the classical Brownian motion starting directly from our non-Markovian
Klein-Kramers and Smoluchowski equations, without alluding to any model
Hamiltonian. Accordingly, quantizing our non-Markovian Klein-Kramers in phase
space gives rise to a non-Markovian quantum master equation in configuration
space, whereas quantizing our non-Markovian Smoluchowski equation in
configuration space leads to a non-Markovian quantum Smoluchowski equation in
phase space. In addition, it is worth noticing that non-Markovian quantum
Brownian motion takes place in presence of a generic environment (e.g. a
non-thermal quantum fluid). As far as the special case of a heat bath
comprising of quantum harmonic oscillators is concerned, a non-Markovian
Caldeira-Leggett master equation and a thermal quantum Smoluchowski equation
are derived and extended to bosonic and fermionic heat baths valid for all
temperatures.
