Dan Asimov wrote:
This MR abstract of a paper by Henry McKean from 1960 looks relevant:
<< MR0133869 (24 #A3693) 60.62 McKean, H. P., Jr. Brownian motions on the 3-dimensional rotation group. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 1960/1961 25–38 . . . An application describes the motion of a sphere rolling without slipping on a plane while its center performs Brownian motion [see also C. D. Gorman, Trans. Amer. Math. Soc. 94 (1960), 103–117; MR0115218 (22 #6020)].
Thanks, Dan! I wasn't able to find that in JSTOR, but I did find Gorman's article, which constructs Brownian motion on the 3-dimensional rotation group precisely by the tactic of rolling a sphere on the plane so that the point of tangency follows a polygonal approximation to a 2-dimensional Brownian motion on that plane; the main result is that you get a well-defined limit as the mesh of the approximation goes to 0. Note, though, that this doesn't answer the question about the graph of a 1-dimensional Brownian motion. Incidentally, I think there is still more to be said about Dan's nice sphere-rolling puzzle:
PUZZLE: Suppose a unit sphere rolls on the plane, once around the unit circle about the origin.
What net spatial rotation (as a 3x3 matrix) does this impart to the sphere?
Gene Salamin submitted a detailed algebraic solution, culminating in the conclusion that the rotation angle is 2 pi sqrt(1 + r^2). But isn't there a purely geometrical solution, where you somehow unroll the motion of the sphere to see a right triangle with legs of length 2 pi and 2 pi r, sort of like the way we measure the length of a geodesic on a cylinder by unrolling it, but different? Here's a start towards such a picture: Stick a spit through the sphere that pierces it at the antipodal points (r+1,0,1) and (r-1,0,1) so that one end of the spit is at (0,0,1). Now spin the spit around (0,0,1), giving a little bit of a twirl as you do. (Think of the way you would have an ox walk in a circle so as to grind wheat at the center of the circle.) When the rolling sphere has completed one circuit, the spit is still going through it, so we can see that computing the axis of net rotation is easy. But how do we see what the rotation angle is? Jim