  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.

Probability