Mean Brownian Paths

Brownian motion is a pure random process and produces paths that are each and very squiggly. In fact so squiggly that their fractal dimension is 2. 

But what about the shape of the path on average?

For any particular section of a random (Brownian) path, we can perform a least squares alignment of every such path and draw the mean path shape:

It is straight because there are as many trajectories that bend anticlockwise as those that bend clockwise. But we are interested in paths rather than trajectories, and the time direction doesn't matter with paths. So here we find the closest Euclidean alignments for either time direction:

It is (I very much suspect) a circular arc. The angle of rotation of this arc would be an interesting constant to discover, it seems to be 1 radian, but that is a rough guess. The bulge is about 0.152 of the arc length, which is interesting because that is exceedingly close to the proposed solution to Moser's worm problem, which is an arc displaced 0.1528 at the centre (An Improved Upper Bound for Leo Moser’s Worm Problem). Moser's worm problem is to find the minimum area shape to accomodate unit length curves, the above curve is the mean unit length curve, because Brownian motion over a unit length is statistically the set of all unit length curves with no bias. So it is interesting that the minimal area shape is potentially a reflected pair of these mean curves:

potential minimum area solution to Moser's worm problem. Bounded by two similar arcs to the mean Brownian path.

The mean curve has bilateral symmetry because we don't allow mirroring in our equivalence class of path shapes. If we do allow mirroring we get:

The two images were for a different number of vertices and different random values, to show that the shape is consistent. This is an average path if you allow flipping, so its mirror image is the other equivalent average path.

You might be wondering what would happen if you don't allow time symmetry but you do allow mirror symmetry. The result appears to be exactly the same as just time symmetry and no mirror symmetry. This makes sense to me. 

Brownian motion in 2D inevitably crosses over itself to form a loop. If we look at just these loops when we get a similar set of results. If time symmetry is disabled then the average circuit is a straight line back-and-forth motion:

If it is enabled then we get a bilaterally symmetric shape:

Here are the points (separate run):

and if we allow mirroring in our equivalence class then we get this funny shape:

and its reflection  (separate run):

We generate these loops because each axis in 2D is performing Brownian motion, which we can model at large scale as a normally distributed offset between each point in turn. In order to get a loop we need each axis to return to the original value, which is called a Brownian bridge. Rather than running a million times until it happens to return to the same value, we simply subtract the linear function from start to finish, which is proven to be equivalent. 

So there we go. Since Brownian motion is a purely random process, these shapes are sure quite relevant as the bulk shape traced by random paths and random loops. We might even think of them as archetypal paths and loops, as they don't seem to contain any prior information, just randomness.

Since there is no prior information in the formulation, the shapes are pretty much definied by the equivalence class used. If you know what your symmetries are when declaring two shapes to be the same, then this dictates what the shape will look like; the result is a function of your interpretation of what a shape is.