TITLE: Quantum analogue of Wiener process
QUESTION [11 upvotes]: The Wiener process (say, on $\mathbb{R}$) can be thought of as a scaling limit of a classical, discrete random walk. On the other hand, one can define and study quantum random walks, when the underlying stochastic process is governed by a unitary transform + measurement (for an excellent introduction, see http://arxiv.org/abs/quant-ph/0303081).
My question is - do quantum random walks have a reasonable continuous limit, something which would give a quantum analogue of the Wiener process?

REPLY [2 votes]: In section III.B of the survey paper you cite, it describes continuous quantum walks, which are I think are a natural analogue of the Wiener process.  These are basically Hamiltonian evolution when the Hamiltonian is something like the adjacency matrix (or Laplacian) of a graph.
On the relationship between continuous- and discrete-time quantum walk has some recent developments with fascinating applications to simulating Hamiltonians on quantum computers.