  The signature of a path \gamma is a sequence whose n-th term is the order-n
iterated integrals of \gamma. It arises from solving multidimensional linear
differential equations driven by \gamma. We are interested in relating the path
properties of \gamma with its signature. If \gamma is C^{1}, then an elegant
formula of Hambly and Lyons relates the length of \gamma to the tail
asymptotics of the signature. We show an analogous formula for the
multidimensional Brownian motion, with the quadratic variation playing a
similar role to the length. In the proof, we study the hyperbolic development
of Brownian motion and also obtain a new subadditive estimate for the
asymptotic of signature, which may be of independent interest. As a corollary,
we strengthen the existing uniqueness results for the signatures of Brownian
motion.
