Under what condition we get back path from signatures in rough path theory?

Question

A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path It appears path and signatures has one to one mapping in many cases. I understand that the signature is not dependent on initial condition and if there is any retraces or double back on itself in the path it cancels out in the signature. Therefore we can never recover the starting point of a path or any segment where the path retraces or doubles back on itself from the signature. The question is given a signature when can we get back the path from which it is derived? It is assumed that the initial condition is zero or known and there is retraces or double back on itself in the entire path (never crosses itself). Any comments would be highly appreciated. Any reference or work in this area would be helpful. Is there a formal procedure to get back the path from the signature?

Answer 1

Loops don't get canceled out in the signature. (You might like to compute the signature of a circle. The second iterated integral is nonzero; in fact, by Green's theorem, it gives you the area inside the circle, maybe up to a factor of $\frac{1}{2}$ or something.) What does get cancelled out is any segment where the path retraces or doubles back on itself. The natural question is which paths have trivial signature.

The idea of signature was essentially introduced by K. T. Chen in 1958. Chen has a notion of "irreducible" path, i.e. one which cannot be written as $\alpha \cdot \gamma \cdot \gamma^{-1} \cdot \beta$ where $\cdot$ is concatenation and $\gamma^{-1}$ is the reversal of $\gamma$. Chen showed that for piecewise regular paths (piecewise $C^1$ with non-vanishing derivative), an irreducible path with trivial signature is trivial. So any path with trivial signature is a finite sequence of "retracings".

Chen, Kuo-Tsai, Integration of paths. A faithful representation of paths by non- commutative formal power series, Trans. Am. Math. Soc. 89, 395-407 (1959). ZBL0097.25803.

More recently, in a very famous paper, Hambly and Lyons considered the case of a bounded variation path. They introduce the notion of a "tree-like path", where in some sense every piece of the path is doubled back; but there might be infinitely many "pieces". They prove that a path of bounded variation has trivial signature iff it is tree-like, and conjectured that this should still hold with less regularity assumptions.

Hambly, Ben; Lyons, Terry, Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. Math. (2) 171, No. 1, 109-167 (2010). ZBL1276.58012.

If you look up papers citing Hambly and Lyons, you can find more recent progress. It appears that "trivial signature iff tree-like", appropriately defined, has been proved for weakly geometric paths.

Boedihardjo, Horatio; Geng, Xi; Lyons, Terry; Yang, Danyu, The signature of a rough path: uniqueness, Adv. Math. 293, 720-737 (2016). ZBL1347.60094.

And for simple paths.

Boedihardjo, Horatio; Ni, Hao; Qian, Zhongmin, Uniqueness of signature for simple curves, J. Funct. Anal. 267, No. 6, 1778-1806 (2014). ZBL1294.60063.

There has also been work on how to explicitly reconstruct a path, up to tree-like equivalence, from its signature.

Geng, Xi, Reconstruction for the signature of a rough path, ZBL06775247.