Non-degeneracy of some Sobolev Pseudo-norms of fractional Brownian motion

Yaozhong Hu (University of Kansas)
Fei Lu (Lawrence Berkeley National Laboratory)
David Nualart (University of Kansas)


Applying an upper bound estimate for $L^{2}$ small ball probability for fractional Brownian motion (fBm), we prove the non degeneracy of some Sobolev pseudo-norms of fBm.

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Pages: 1-8

Publication Date: November 3, 2013

DOI: 10.1214/ECP.v18-2986


  • Airault, H.; Malliavin, P. Intégration géométrique sur l'espace de Wiener. (French) [Geometric integration on the Wiener space] Bull. Sci. Math. (2) 112 (1988), no. 1, 3--52. MR0942797
  • Bronski, Jared C. Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions. J. Theoret. Probab. 16 (2003), no. 1, 87--100. MR1956822
  • Dzhaparidze, Kacha; van Zanten, Harry. Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion. Ann. Probab. 33 (2005), no. 2, 620--644. MR2123205
  • Fang, Shizan. Non-dégénérescence des pseudo-normes de Sobolev sur l'espace de Wiener. (French) [Nondegeneracy of Sobolev pseudonorms on Wiener space] Bull. Sci. Math. 115 (1991), no. 2, 223--234. MR1101025
  • Li, Wenbo V.; Shao, Qi-Man. Small ball estimates for Gaussian processes under Sobolev type norms. J. Theoret. Probab. 12 (1999), no. 3, 699--720. MR1702899
  • Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, 533--597, Handbook of Statist., 19, North-Holland, Amsterdam, 2001. MR1861734
  • Malliavin, Paul. $C^{k}$-hypoellipticity with degeneracy. II. Stochastic analysis (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1978), pp. 327--340, Academic Press, New York-London, 1978. MR0517250
  • Nazarov, A. I.; Nikitin, Ya. Yu. Logarithmic asymptotics of small deviations in the $L_ 2$-norm for some fractional Gaussian processes. (Russian) Teor. Veroyatn. Primen. 49 (2004), no. 4, 695--711; translation in Theory Probab. Appl. 49 (2005), no. 4, 645--658 MR2142562
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5 MR2200233

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