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In the present paper, we are going to show that outside a slim set in thesense of Malliavin (or quasi-surely), the signature path (which consists ofiterated path integrals in every degree) of Brownian motion isnon-self-intersecting. This property relates closely to a non-degeneracyproperty for the Brownian rough path arising naturally from the uniqueness ofsignature problem in rough path theory. As an important consequence we concludethat quasi-surely, the Brownian rough path does not have any tree-like piecesand every sample path of Brownian motion is uniquely determined by itssignature up to reparametrization.

A Quasi-sure Non-degeneracy Property for the Brownian Rough Path