Can quantum Brownian motion be Markovian?

The question whether a dissipative quantum system can approximately be described by a Markov process is reexamined using Brownian motion in a field of force as an example. The exactly solvable model of a damped harmonic oscillator allows for a precise identification of the range of validity of Markovian evolution equations. Such equations are valid both in the limits of weak and strong damping if the temperature is not too low. From these findings the parameter domain within which the weak coupling quantum master equation and the strong coupling quantum Smoluchowski equation can be employed to describe nonlinear quantum systems is outlined.