Tail estimates for Markovian rough paths

Abstract

The accumulated local $p$-variation functional [Ann. Probab. 41 (213) 3026–3050] arises naturally in the theory of rough paths in estimates both for solutions to rough differential equations (RDEs), and for the higher-order terms of the signature (or Lyons lift). In stochastic examples, it has been observed that the tails of the accumulated local $p$-variation functional typically decay much faster than the tails of classical $p$-variation. This observation has been decisive, for example, for problems involving Malliavin calculus for Gaussian rough paths [Ann. Probab. 43 (2015) 188–239].

All of the examples treated so far have been in this Gaussian setting that contains a great deal of additional structure. In this paper, we work in the context of Markov processes on a locally compact Polish space $E$, which are associated to a class of Dirichlet forms. In this general framework, we first prove a better-than-exponential tail estimate for the accumulated local $p$-variation functional derived from the intrinsic metric of this Dirichlet form. By then specialising to a class of Dirichlet forms on the step $\lfloor p\rfloor $ free nilpotent group, which are sub-elliptic in the sense of Fefferman–Phong, we derive a better than exponential tail estimate for a class of Markovian rough paths. This class includes the examples studied in [Probab. Theory Related Fields 142 (2008) 475–523]. We comment on the significance of these estimates to recent papers, including the results of Ni Hao [Personal communication (2014)] and Chevyrev and Lyons [Ann. Probab. To appear].