A generalized Fernique theorem and applications
Authors:
Peter Friz and Harald Oberhauser
Journal:
Proc. Amer. Math. Soc. 138 (2010), 36793688
MSC (2010):
Primary 60G15, 60H99; Secondary 60B99
Published electronically:
June 15, 2010
MathSciNet review:
2661566
Fulltext PDF Free Access
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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):546578, 2002] then all follow from applying our main theorem.
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 1.
 T. Cass, P. Friz, and N. Victoir.
Nondegeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc., 361:33593371, 2009.
 2.
 A. M. Davie.
Differential equations driven by rough signals: An approach via discrete approximation. Applied Mathematics Research eXpress, 2007.
 3.
 P. Friz.
Continuity of the Itômap for Hölder rough paths with applications to the support theorem in Hölder norm. In Probability and partial differential equations in modern applied mathematics, volume 140 of IMA Vol. Math. Appl., pages 117135. Springer, New York, 2005. MR 2202036 (2007f:60070)
 4.
 P. Friz and N. Victoir.
Approximations of the Brownian rough path with applications to stochastic analysis. Ann. Inst. H. Poincaré Probab. Statist., 41(4):703724, 2005. MR 2144230 (2007e:60018)
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 P. Friz and N. Victoir.
A note on the notion of geometric rough paths. Probab. Theory Related Fields, 136(3):395416, 2006. MR 2257130 (2007k:60114)
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Differential equations driven by Gaussian signals. Ann. Inst. H. Poincaré (B) Probab. Statist. 46(2): 369413, 2010.
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 Yuzuru Inahama.
Laplace's method for the laws of heat processes on loop spaces. Journal of Functional Analysis, 232(1):148  194, 2006. MR 2200170 (2007g:60060)
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Isoperimetry and Gaussian analysis. In Lectures on probability theory and statistics (SaintFlour, 1994), volume 1648 of Lecture Notes in Math., pages 165294. Springer, Berlin, 1996. MR 1600888 (99h:60002)
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Lévy area of Wiener processes in Banach spaces. Ann. Probab., 30(2):546578, 2002. MR 1905851 (2003h:60088)
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System Control and Rough Paths. Oxford Mathematical Monographs, Oxford University Press, 2002. MR 2036784 (2005f:93001)
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 S. J. Taylor.
Exact asymptotic estimates of Brownian path variation. Duke Math. J., 39:219241, 1972. MR 0295434 (45:4500)
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Additional Information
Peter Friz
Affiliation:
Institut für Mathematik, Technical University of Berlin, D10623 Berlin, Germany – and – Weierstrass Institut for Angewandte Analysis and Stochastik, Berlin, Germany
Email:
friz@math.tuberlin.de
Harald Oberhauser
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, England
Email:
h.oberhauser@statslab.cam.ac.uk
DOI:
http://dx.doi.org/10.1090/S000299392010105282
Received by editor(s):
September 29, 2009
Published electronically:
June 15, 2010
Additional Notes:
The second author was supported by EPSCR Grant EP/P502365/1 and a DOCfellowship of the Austrian Academy of Sciences
Communicated by:
Peter A. Clarkson
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
