The only rough path that I've ever seen discussed are the ones associated with Brownian motion. I could use a "rough path" for any nice function, defeating the point. In particular are there interesting rough paths that are $\alpha$-Holder continuous for $\frac13 \lt \alpha \lt \frac12$?



What's an example of a rough path that's not Ito/Stratonovich-Brownian rough path?
[An explicit rough path construction for continuous paths with arbitrary Hölder exponent](https://hal.inria.fr/hal-00370570/document), by Jeremie Unterberger (2009) [[published](http://link.springer.com/article/10.1007%2Fs00220-010-1064-1) with a different title]:



> 
> We construct an explicit geometric rough path over arbitrary
>  $d$-dimensional paths with finite $1/\alpha$-variation for any
>  $\alpha\in (0, 1)$. The method generalizes to arbitrary $\alpha$-Hölder paths the previous constructions that were limited to fractional Brownian motion. Our generalization may
>  be coined as *Fourier normal ordering* since it consists in a
>  regularization obtained after permuting the order of integration in
>  iterated integrals so that innermost integrals have highest Fourier
>  frequencies. The method is not limited to Brownian motion, because it does not rely on tools belonging exclusively to the Gaussian realm, namely, the equivalence of $L^p$-norms due to the hypercontractivity property of the Ornstein-Uhlenbeck process.
> 
> 
> 


For a broader overview of this topic with many examples, starting from T. Lyon's seminal contributions, see Unterberger's [Rough Path Theory](http://www.iecn.u-nancy.fr/~unterber/book-rough-paths.pdf).


