  Based on the dynamical quantization method we derive a quantum phase-space
non-Markovian Smoluchowski equation describing the non-inertial Brownian motion
of a harmonic oscillator immersed in a generic environment. In the long-time
regime we investigate the tunneling phenomenon by evaluating the quantum
Kramers escape rate of a Brownian particle over a potential barrier. As far as
a quantum thermal reservoir is concerned, it is found that our steady quantum
Kramers rate may depend upon no friction constant in the strong friction
domain. This preposterous feature may suggest the existence of non-dissipative
quantum tunneling at the low-temperature range, including the zero temperature
case. Lastly, we predict that our quantum escape rate non-exponentially decays
on the edge of the breakdown of the dissipation-fluctuation relation.
