  On the basis of the dynamical-quantization approach to open quantum systems,
we can derive a non-Markovian Caldeira-Leggett quantum master equation as well
as a non-Markovian quantum Smoluchowski equation in phase space. On the one
hand, we solve our Caldeira-Leggett equation for the case of a quantum Brownian
particle in a gravitational field. On the other hand, we solve our quantum
Smoluchowski equation for a harmonic oscillator. In both physical situations we
come up with the existence of a non-equilibrium thermal quantum force. Further,
as a physical application of our quantum Smoluchowski equation we take up the
phenomenon of escape rate of a non-inertial Brownian particle over a potential
barrier.
  Key-words: Quantum Brownian motion; Non-Markovian effects; Caldeira-Leggett
quantum master equation; Quantum Smoluchowski equation; Quantum tunneling
