analysis / Rough Paths, Path Signatures
For a continuous bounded-variation path with signature $g$, logarithmic signature $l$ and increment $v$, the modified Lyons–Sidorova conjecture predicts the structure of $g$ when $R(l)=\infty$. The paper proves it: $g=1$ when $v=0$, and otherwise a prefix $\alpha$ of the centred path gives $S(\gamma) = S(\alpha)e^{v}S(\alpha)^{-1}$.
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This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.
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For a continuous bounded-variation path with signature $g$, logarithmic signature $l$ and increment $v$, the modified Lyons–Sidorova conjecture predicts the structure of $g$ when $R(l)=\infty$. The paper proves it: $g=1$ when $v=0$, and otherwise a prefix $\alpha$ of the centred path gives $S(\gamma) = S(\alpha)e^{v}S(\alpha)^{-1}$.
This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.
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