Probab. Theory Relat. Fields (2018) 170:891–932
Random walks and Lévy processes as rough paths
Received: 9 November 2015 / Revised: 22 February 2017 / Published online: 17 May 2017
© The Author(s) 2017. This article is an open access publication
Abstract We consider random walks and Lévy processes in a homogeneous group
G. For all p > 0, we completely characterise (almost) all G-valued Lévy 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 Lévy process in p-variation
topology. In the case that G is the free nilpotent Lie group over R
, 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 Lévy–Khintchine
formula for the characteristic function of the signature of a Lévy process. At the heart
of our analysis is a criterion for tightness of p-variation for a collection of càdlàg
strong Markov processes.
Keywords Homogeneous groups · Rough paths · Lévy processes · Random walks ·
Tightness of p-variation · Stochastic ﬂows · Characteristic functions of signatures
Mathematics Subject Classiﬁcation Primary 60G51; Secondary 60H10
This paper focuses on several questions regarding Lévy processes and random walks
in homogeneous groups, with a particular focus on applications to rough paths theory.
The author is supported by a Junior Research Fellowship of St John’s College, Oxford.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory
Quarter, Woodstock Road, Oxford OX2 6GG, UK