Sanity check to see if I understand correctly: dx is a (smooth) function from $\Bbb R^3$ to its cotagent bundle. Pointwise it induces functionals on the cotagent spaces. (In general a $k$-form is a function from your manifold to the bundle of $k$-forms and it induces a $k$-multilinear alternating map on each $(T_pM)^k$)
Given a connection on a bundle (when M is a Riemannian manifold, the connection I just talked about is on TM), a section of E is 'parallel' if the connection takes sigma(p) to sigma(q) whenever you parallel translate from p to q.
If you think of R^3 as a Riemannian manifold, the fact that dx seems constant is not illusory: it is because dx is a parallel 1-form.
In Italy while everyone was going crazy about palm oil, in full public outrage, Ferrero was like "we don't give a damn, we can put kerosene into nutella and people will still buy it" so they made a TV commercial in which they explicitely say that all of their ingredients are good, including palm oil
Oh, that's a serious journal. I suspect they are broke. Probably most journals are, because university libraries have cut back budgets to pay for journals.
OK, @EricSilva. I'll give it a try when you cook it for me.
Well, @Alessandro, I actually meant doing some pullbacks and computing a few integrals explicitly. This is the Frobenius Theorem, which is super important.
So the point is, for normal integration, (for example Riemann-Stieltjes integral) the pathwise integral $\int \phi(dx)$ is not continuous for finite $p$-variation.
What rough path theory tells you is, the missing information, which is need to converge, is contained in the signature of path. Signature of the path is just a series of iterated integral.
In mathematical manner is, you can see the signature as a path in the Lie group. And on the space of the Lie group you can find a sub riemannian structure. And the geodesic is just the signature of a bounded variation path.
@TedShifrin Yeah not parametrix. I remember it was mentioned in my notes somewhere. It has an equation that that resembles something like $\sqrt{p(x)-\epsilon^2}$ where p is the potential function.
I still don't understand the meaning of the word. It of course has nothing to do with other uses of "signature" in mathematics (relating to quadratic forms).
@TedShifrin OMG! It was separatrix! I don't know why I was sure that wasn't it. The equation that I was misremembering is $p = \pm \frac{q}{\sqrt{2}}\sqrt{2-q^2}$.
The idea with the hölder norm is to quantify how "rough" your path is. The more "rough" your path is, the "smooth" your 1-form should be to kill the roughness in your path.
i thought its a very small community who does rough path.
