Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard Riemann-Stieltjes integrals $\mathbb X_{s,t}^n=\int_s^t (X^n(r)-X^n(s) )\otimes dX^n(r)$. Then is it true that $\sup_n\|\mathbb X^n\|_{2\alpha}<\infty$?

This is related to rough paths theory and the interpolation result of Friz and Hairer's exercise 2.9. I am wondering if the second condition really needs to be checked.


1 Answer 1


This cannot hold, and in a sense rough path theory has to be developed precisely because of this reason; otherwise, rough path lifts would be defined uniquely for any curve of Hölder regularity $>1/3$.

For a simple example, take $Y_n:t\mapsto n^{-1}\exp(in^2t)$, which converges to zero in every $(1/2-\varepsilon)$-Hölder norm (for instance by looking at times $|t-s|\geq1/n^2$ or $|t-s|\leq1/n^2$). The component of $\mathbb Y_{0,1}$ in $1\otimes i$ converges to $1/2$, so $X_n=Y_n/\|Y_n\|_\alpha$ is a counterexample for all $\alpha<1/2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.