TITLE: Could someone explain rough path theory? More specifically, what is the higher ordered "area process" and what information is it giving us?
QUESTION [24 upvotes]: http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, $(X,\mathbb{X})$ where $X$ is a continuous process and $\mathbb{X}$ is a higher ordered area process which defines certain integrals that satisfy "Chen's Relation". What is this area process and what does it have to do with the original continuous function? What are these integrals (which he emphasizes are defined by the area process, not the other way around)? So could the area process be anything satisfying Chen's relations?

REPLY [4 votes]: Consider an equation like
\begin{equation}\tag{1}dY_t = f(Y_t) dW_t\end{equation}
where $Y_t$ is an unknown function and $W_t$ is a continuous, but not differentiable, function.

If $W_t$ is Brownian motion then there is a classical theory for how to understand (1). Brownian motion has finite $p$-variation only for certain values of $p$. As you may know, Brownian motion on the interval $[0,T]$ has quadratic variation equal to $T$.
What if $W_t$ is not Brownian motion, but just some function with finite $p$-variation for some other $p$?
Then the approach is to just fix a path of $W_t$ (so it's not a random process anymore, just a single non-differentiable function). Then Lyons shows how to make sense of (1) in terms of
$$\tag{2}\int W_s dW_s$$
We don't know what (2) means, so we write down some axioms for what (2) should be like (those are Chen's relations I guess), and use that as a starting point for the theory.

(This is based on my reading of the first page of http://www.maths.ed.ac.uk/~adavie/rpathrev.pdf)