Abstract
In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-self-intersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization.
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Research supported partly by the ERC grant (Grant Agreement No. 291244 ESig).
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Boedihardjo, H., Geng, X., Liu, X. et al. A Quasi-sure Non-degeneracy Property for the Brownian Rough Path. Potential Anal 51, 1–21 (2019). https://doi.org/10.1007/s11118-018-9699-1
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DOI: https://doi.org/10.1007/s11118-018-9699-1