A quasi-sure non-degeneracy property for the Brownian rough path (Q2312624)

This paper is concerned with the non-degeneracy of the \(d\)-dimensional Brownian path in the sense of the absence of self-intersections in its signature, constructed as a series of iterated path integrals. The authors consider the truncation of the signature of \(d\)-dimensional Brownian up to iterated integral degree \(n\geq 1\), and show that that the set of Brownian paths with non-intersecting truncated signature has \((r,q)\)-capacity zero in the Malliavin sense, provided that \(\binom{n+d-1}{n} > rq+4\). This result can be interpreted as a quasi-sure absence of tree-like pieces in the Brownian rough path and shows that, up to reparametrization, the sample paths of Brownian motion are quasi-surely uniquely determined by their signature.