A generalized Fernique theorem and applications

Author:

Friz Peter,Oberhauser Harald

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 Hölder norms of the (fractional) Brownian rough path, Gaussian rough paths and the Banach space valued Wiener process enhanced with its Lévy area [Ledoux, Lyons, Qian. “Lévy area of Wiener processes in Banach spaces”, Ann. Probab., 30(2):546–578, 2002] then all follow from applying our main theorem.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference12 articles.

1. T. Cass, P. Friz, and N. Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc., 361:3359–3371, 2009.

2. A. M. Davie. Differential equations driven by rough signals: An approach via discrete approximation. Applied Mathematics Research eXpress, 2007.

3. Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm;Friz, Peter K.,2005

4. Approximations of the Brownian rough path with applications to stochastic analysis;Friz, Peter;Ann. Inst. H. Poincar\'{e} Probab. Statist.,2005

5. A note on the notion of geometric rough paths;Friz, Peter;Probab. Theory Related Fields,2006

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