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

Dynamical quantization of the non-Markovian Smoluchowski equation and the quantum tunneling phenomenon