Quasi-sure Existence of Gaussian Rough Paths and Large Deviation Principles for Capacities

TL;DR

This paper constructs a quasi-sure framework for Gaussian rough paths with long memory and establishes large deviation principles for capacities, advancing the understanding of stochastic differential equations driven by such processes.

Contribution

It introduces a quasi-sure construction of Gaussian rough paths with long-time memory and proves a large deviation principle for capacities, extending existing theories.

Findings

Quasi-sure construction of Gaussian rough paths with long memory

Large deviation principles for capacities of these paths

Immediate implications for SDE solutions driven by Gaussian processes

Abstract

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.