## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 4 (2017), 2477-2504.

### Tail estimates for Markovian rough paths

Thomas Cass and Marcel Ogrodnik

#### 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].

#### Article information

**Source**

Ann. Probab., Volume 45, Number 4 (2017), 2477-2504.

**Dates**

Received: December 2014

Revised: September 2015

First available in Project Euclid: 11 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1502438432

**Digital Object Identifier**

doi:10.1214/16-AOP1117

**Mathematical Reviews number (MathSciNet)**

MR3693967

**Zentralblatt MATH identifier**

06786086

**Subjects**

Primary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G20: Generalized stochastic processes 65C30: Stochastic differential and integral equations

**Keywords**

Rough path theory Dirichlet form Markov process tail estimates

#### Citation

Cass, Thomas; Ogrodnik, Marcel. Tail estimates for Markovian rough paths. Ann. Probab. 45 (2017), no. 4, 2477--2504. doi:10.1214/16-AOP1117. https://projecteuclid.org/euclid.aop/1502438432