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The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths

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modified-lyons-sidorova-conjectureAnalysisposed by Horatio Boedihardjo; formulated systematically by Boedihardjo, Geng and Wang, after the original conjecture of Lyons and Sidorova, 2020recorded: solved

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Statement

For a continuous bounded-variation path with signature gg, logarithmic signature ll and increment vv, the modified Lyons–Sidorova conjecture predicts the structure of gg when R(l)=∞R(l)=\infty. The paper proves it: g=1g=1 when v=0v=0, and otherwise a prefix α\alpha of the centred path gives S(γ)=S(α)evS(α)−1S(\gamma) = S(\alpha)e^{v}S(\alpha)^{-1}.

Context

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.

A named conjecture of Lyons and Sidorova about when a path signature determines the path up to conjugacy, in the setting of bounded variation.

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  • #1

    Attempt 1

    proof attemptChatGPT with Elena Boguslavskaya ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
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    ChatGPT
    people
    Elena Boguslavskaya

    The author reports using ChatGPT for mathematical exploration, critical examination of arguments, testing of intermediate proof strategies, and drafting and editing, and states she reviewed and revised all AI-assisted material and takes full responsibility. The model is not credited with producing the proof.

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