Transportation–cost inequalities for diffusions driven by Gaussian processes

Sebastian Riedel

doi:10.1214/17-ejp40

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 Frechet 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 [FH14, Theorem 11.7] used in rough paths theory.

Highlights

Transportation–cost inequalities can be seen as a functional approach to the concentration of measure phenomenon
We say that a p-transportation–cost inequality holds for a measure μ ∈ P (E) if there is a constant C such that
We further study the case where b only satisfies a one-sided Lipschitz condition in Theorem 2.6

Summary

Introduction

Transportation–cost inequalities can be seen as a functional approach to the concentration of measure phenomenon (cf. Ledoux’s work [Led01] for an introduction to the theory of measure concentration and the work [GL10] by Gozlan and Leonard for an overview to transport inequalities). In the case when X is a Brownian motion, the equation (0.2) can be solved using Ito’s framework, and transportation–cost inequalities were studied in many works: In this context, T1(C) was first established for the law of Y with respect to the uniform metric by Djellout, Guillin and Wu in [DGW04]. Together with our result about transport inequalities on Gaussian spaces, all we need to establish is Lipschitzness of the solution map X(ω) → Y (ω) for equation (0.2) This is usually true for additive noise, and we study this case in Section 2.1 first. If B is a Banach space, C0p−var([S, T ]; B) is a Banach space with the norm · p−var

Transportation inequality on a Gaussian space

Applications to diffusions

Tail estimates for functionals