A generalized Fernique theorem and applications

TL;DR

This paper extends Fernique's theorem to a broader class of functionals on Wiener spaces using isoperimetric inequalities, enabling Gaussian integrability results for complex stochastic processes like rough paths.

Contribution

It introduces a generalized Fernique theorem applicable to non-Gaussian functionals on Wiener spaces, with applications to rough path theory and Gaussian processes.

Findings

Established Gaussian integrability for rough path norms.

Derived explicit constants for fractional Brownian rough paths.

Extended Fernique's theorem to a wider class of functionals.

Abstract

We prove a generalisation of Fernique's theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough path theory where one deals with iterated integrals of Gaussian processes (which are generically not Gaussian). Gaussian integrability with explicitly given constants for variation and H\"older norms of the (fractional) Brownian rough path, Gaussian rough paths and the Banach space valued Wiener process enhanced with its L\'evy area [Ledoux, Lyons, Quian. "L\'evy area of Wiener processes in Banach spaces". Ann. Probab., 30(2):546--578, 2002] then all follow from applying our main theorem.