Transportation-cost inequalities for diffusions driven by Gaussian processes

TL;DR

This paper establishes transportation-cost inequalities for solutions of SDEs driven by Gaussian processes, including fractional Brownian motion, using rough paths theory, and connects these inequalities to Gaussian tail estimates.

Contribution

It introduces new transportation-cost inequalities for Gaussian-driven SDEs and provides a novel proof of Talagrand's inequality on Gaussian Fréchet spaces, extending previous results.

Findings

Transportation-cost inequalities hold for Gaussian process-driven SDEs.

A new proof of Talagrand's inequality on Gaussian Fréchet spaces is provided.

Establishing transportation-cost inequalities simplifies proving Gaussian tail estimates.

Abstract

We prove transportation-cost inequalities for the law of SDE solutions driven by general Gaussian processes. Examples include the fractional Brownian motion, but also more general processes like bifractional Brownian motion. In case of multiplicative noise, our main tool is Lyons' rough paths theory. We also give a new proof of Talagrand's transportation-cost inequality on Gaussian Fr\'echet spaces. We finally show that establishing transportation-cost inequalities implies that there is an easy criterion for proving Gaussian tail estimates for functions defined on that space. This result can be seen as a further generalization of the "generalized Fernique theorem" on Gaussian spaces [Friz-Hairer 2014; Theorem 11.7] used in rough paths theory.