I've read a fairly extensive amount of Wiener's work and he never defined Brownian motion as a random walk to my knowledge. Replace "walk" with "process" and you got it. Actually, you need to go way back to the joint work of Paley and Wiener and their work on the "fundamental random function" which is continuous almost everywhere but not differentiable in the usual sense. Such things are constructed, not merely hypothesized to exist, it's fairly involved and technical. Also, nearly all of Wiener's stuff(that I've read) was very applied and practical due to his work on automatic fire control(guns.. a lot of the stuff was originally classified), cybernetics, radar systems, tracking, etc. Also, individual points on this manifold/space/whatever are not described in detail, but counts of particles in boxes/intervals/cubes/hypercubes/etc of varying sizes are, and then you can define in-flow out-flow, "weak directional derivatives", for each of these boxes into each other and calculate the evolution deterministically. The "randomness" only appears in the actual sampling of the process and that goes to inference theory. Actually, now that I think about it, the density of the dust particles in space are defined by the parameters of the densities in each of these successively refined and nested boxes where the size of the boxes approaches 0 as the number of boxes approaches infinity. Some connection to Ricci tensors? E.g. uniform density would be Euclidean and space would be "flat" but a non-uniform density of dust particles could be described by some analog of a "stochastic ricci tensor" since the BM is only weakly differentiable.. actually, this looks interesting, only briefly glanced. http://arxiv.org/pdf/hep-th/9812254For
all intents and purposes, BM just means gaussian distribution with variance given by |t-s| and the implications of such a process. Anything else besides that definition is not something I pay attention to. It's also well known that the BM does not define any financial series that I'm aware of either, but it does provide a very useful tool to project other more complicated processes onto for algorithmic/computation purposes.