Random walks and L\'evy processes as rough paths

We consider random walks and L'evy processes in a homogeneous group $G$. For all $p>0$, we completely characterise (almost) all $G$-valued L'evy processes whose sample paths have finite $p$-variation, and give sufficient conditions under which a sequence of $G$-valued random walks converges in law to a L'evy process in $p$-variation topology. In the case that $G$ is the free nilpotent Lie group over ${\mathbb{R}}^d$, so that processes of finite $p$-variation are identified with rough paths, we demonstrate applications of our results to weak convergence of stochastic flows and provide a L'evy-Khintchine formula for the characteristic function of the signature of a L'evy process. At the heart of our analysis is a criterion for tightness of $p$-variation for a collection of c`adl`ag strong Markov processes.