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.

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I understand the question is more oriented towards the physical phenomenon of Brownian motion, which I would describe as the effect of an environment with a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. We would then be seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of <A HREF="http://en.wikipedia.org/wiki/Quantum_dissipation">Caldeira and Leggett.</A> The literature is very extensive, an older but still relevant review is <A HREF="http://www.researchgate.net/publication/222287864_Quantum_Brownian_motion_The_functional_integral_approach">Quantum Brownian Motion: The Functional Integral Approach</A>.

> The quantum mechanical dynamics of a particle coupled to a heat bath
> is treated by functional integral methods and a generalization of the
> Feynman-Vernon influence functional is derived. The extended theory
> describes the time evolution of nonfactorizing initial states and of
> equilibrium correlation functions. The theory is illuminated through
> exactly solvable models.