To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the *quantum random walk*.

For a physics perspective, see <A HREF="http://arxiv.org/abs/quant-ph/0303081">Quantum random walks - an introductory overview</A>, but you might prefer the more math-oriented exposition of <A HREF="http://arxiv.org/abs/math/0211356">Martin boundary theory of some quantum random walks</A> and <A HREF="http://arxiv.org/abs/quant-ph/0510128">On algebraic and quantum random walks</A>.

> We give a concise prescription of the concept of a quantum random walk
> (QRW), using the example of QRW on integers as paradigm. It briefly
> explains the notion of quantum coin system and the coin tossing map,
> and summarizes two emblematic properties of that walk, namely the
> quadratic enhancement of its diffusion rate due to quantum
> entanglement between the walker and the entropy increase without
> majorization effect of its probability distributions. We conclude with
> a group theoretical scheme of classification of various known QRW's.