Title:

The signature of a rough path : uniqueness

The main contribution of the present thesis is in two aspects. The first one, which is the heart of the thesis, is to explore the fundamental relation between rough paths and their signatures. Our main goal is to give a geometric characterization of the kernel of the signature map in different situations. In Chapter Two, we start by establishing a general fact that a continuous Jordan curve on a Riemannian manifold can be arbitrarily well approximated by piecewise minimizing geodesic interpolations which are again Jordan. This result enables us to prove a generalized version of Green’s theorem for planar Jordan curves with finite pvariation 1 ≤ p < 2, and to prove that two such Jordan curves have the same signature if and only if they are equal up to reparametrization. In Chapter Three, we investigate the problem for general weakly geometric rough paths. In particular, we show that a weakly geometric rough path has trivial signature if and only if it is treelike in the sense we will define later on. In Chapter Four, we study the problem in the probabilistic setting. In particular, we show that for a class of stochastic processes, with probability one the sample paths are determined by their signatures up to reparametrization. A fundamental example is Gaussian processes including fractional Brownian motion with Hurst parameter H > 1/4, the OrnsteinUhlenbeck process and the Brownian bridge. The second one is an application of rough path theory to the study of nonlinear diffusions on manifolds under the framework of nonlinear expectations. In Chapter Five, we begin by studying the geometric rough path nature of GBrownian motion. This enables us to introduce rough differential equations driven by GBrownian motion from a pathwise point of view. Next we establish the fundamental relation between rough (pathwise theory) and stochastic (L^{2}theory) differential equations driven by GBrownian motion. This is a crucial point of understanding nonlinear diffusions and their generating heat flows on manifolds from an intrinsic point of view. Finally, from the pathwise point of view we construct GBrownian motion on a compact Riemannian manifold and establish its generating heat flow for a class of Gfunctions under orthogonal invariance. As an independent interest, we also develop the EulerMaruyama scheme for stochastic differential equations driven by GBrownian motion.
