Characteristic functions of path signatures and applications

The main object of study in this work is the extension of the classical characteristic function to the setting of path signatures. Our first fundamental result exhibits the following geometric interpretation: the path signature is completely determined by the development of the path into compact Lie groups. This faithful representation of the signature is the primary tool we use to define and study the characteristic function. Our investigation of the characteristic function can be divided into two parts. First, we employ the characteristic function to study the expected signature of a path as the natural generalisation of the moments of a real random variable. In this direction, we provide a solution to the moment problem, and study analyticity properties of the characteristic function. In particular, we solve the moment problem for signatures arising from families of Gaussian and Markovian rough paths. Second, we study the characteristic function in relation to the solution map of a rough differential equation. The connection stems from the fact that the signature of a geometric rough path completely determines the path's role as a driving signal. As an application, we demonstrate that the characteristic function can be used to determine weak convergence of flows arising from rough differential equations. Along the way, we develop tools to study càdlàg processes as rough paths and to determine tightness in p-variation topologies of random walks. As a consequence, we provide a classification of Lévy processes possessing sample paths of finite p-variation and determine a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process.