Rough path explorer

Move the slider to change the regularity α of a random path. Close to 1 the path is almost smooth; close to 0 it is wildly rough, which is the regime where rough path theory is needed to make sense of differential equations driven by it.

What you are seeing

Each curve is a sample of fractional Brownian motion with Hurst parameter H = α, generated exactly from its covariance matrix. Such a path is (almost) α-Hölder continuous:

The smaller α is, the faster the path can oscillate on small time scales. When changing α the random numbers stay the same, so you can watch the same path become rougher or smoother. In the plane view, two independent paths are drawn against each other, coloured from start (blue) to end (orange); this is how paths driving a rough differential equation are usually pictured.

For α > 1/2 classical tools still work; for α ≤ 1/3 one needs higher-order information about the path. My paper with Alberto Bonicelli and Lorenzo Zambotti proves a priori bounds for rough differential equations over the full range α ∈ (0,1). More on the blog and on the home page.