The Annals of Probability

Lévy area of Wiener processes in Banach spaces

M. Ledoux, T. Lyons, and Z. Qian

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The goal of this paper is to construct canonical Lévy area processes for Banach space valued Brownian motions via dyadic approximations. The significance of the existence of canonical Lévy area processes is that a (stochastic) integration theory can be established for such Brownian motions (in Banach spaces). Existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of Lyons and Lyons and Qian. This investigation involves a careful analysis on the choice of tensor norms, motivated by the applications to infinite dimensional stochastic differential equations.

Article information

Ann. Probab., Volume 30, Number 2 (2002), 546-578.

First available in Project Euclid: 7 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H15: Stochastic partial differential equations [See also 35R60] 60J60: Diffusion processes [See also 58J65] 60G15: Gaussian processes

Brownian motion differential equation Gaussian comparison theorem Gaussian measure rough path


Ledoux, M.; Lyons, T.; Qian, Z. Lévy area of Wiener processes in Banach spaces. Ann. Probab. 30 (2002), no. 2, 546--578. doi:10.1214/aop/1023481002.

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  • [1] ADLER, R. J. (1990). An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA.
  • [2] BASS, R. F., HAMBLY, B. M. and LYONS, T. J. (1999). Extending the Wong-Zakai theorem to reversible Markov processes. Preprint.
  • [3] BRZE ´ZNIAK,(1997). On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61 245-295.
  • [4] BRZE ´ZNIAK,and CARROLL, A. (2000). Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces. Preprint.
  • [5] BRZE ´ZNIAK,and ELWORTHY, D. (1999). Stochastic differential equations on Banach manifolds. Preprint.
  • [6] CAPITAINE, M. and DONATI-MARTIN, C. (2001). The Lévy area for the free Brownian motion. J. Funct. Anal. 179 153-169.
  • [7] CHEVET, S. (1977). Un résultat sur les mesures gaussiennes. C. R. Acad. Sci. Paris Sér. I Math. 284 441-444.
  • [8] COUTIN, L. and QIAN,(2002). Stochastic analysis, rough path analysis and fractional Brownian motion. Probab. Theory Related Fields 122 108-140.
  • [9] DA PRATO, G. and ZABCZYK, J. (1992). Stochastic equations in infinite dimensions. In Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press.
  • [10] DUDLEY, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 290-330.
  • [11] FERNIQUE, X. (1970). Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris Sér. I Math. 270 1698-1699.
  • [12] FERNIQUE, X. (1975). Regularité des trajectoires des fontions aléatoires gaussiennes. Ecole d'ete probabilites de St. Flour IV. Lecture Notes in Math. 480 1-96. Springer, New York.
  • [13] GOODMAN, V. and KUELBS, J. (1991). Rates of clustering for some Gaussian self-similar processes. Probab. Theory Related Fields 88 47-75.
  • [14] GROSS, L. (1965). Abstract Wiener spaces. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 31-42. Univ. California Press, Berkeley.
  • [15] GROSS, L. (1970). Lecture on Modern Analysis and Applications II. Lecture Notes in Math. 140. Springer, New York.
  • [16] HAMBLY, B. M. and LYONS, T. J. (1998). Stochastic area for Brownian motion on Sierpinski gasket. Ann. Probab. 26 132-148.
  • [17] KUO, H.-H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Math. 436. Springer, New York.
  • [18] LANDAU, H. J. and SHEPP, L. A. (1970). On the supremum of a Gaussian process. Sankhya Ser. A 32 369-378.
  • [19] LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Springer, New York.
  • [20] LEDOUX, M. (1996). Isoperimetry and Gaussian Analysis. Ecole d'Eté de Probabilités de St. Flour. Lecture Notes in Math. 1648 165-294. Springer, New York.
  • [21] LI, W. and LINDE, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556-1578.
  • [22] LIFSHITS, M. A. (1995). Gaussian Random Functions. Kluwer, Dordrecht.
  • [23] LYONS, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215-310.
  • [24] LYONS, T. J. and QIAN,(1998). Flows of diffeomorphisms induced by a geometric multiplicative functionals. Probab. Theory Related Fields 112 91-119.
  • [25] LYONS, T. J. and QIAN,(2000). System Control and Rough Paths. Oxford Univ. Press.
  • [26] SIPPILAINEN, E.-M. (1993). A pathwise view of solutions of stochastic differential equations. Ph.D. dissertation, Univ. Edinburgh.
  • [27] STOLZ, W. (1993). Une méthode élémentaire pour l'évaluation de petites boules browniennes. C. R. Acad. Sci. Paris Sér. I Math. 316 1217-1220.
  • [28] STOLZ, W. (1996). Some small ball probabilities for Gaussian processes under non-uniform norms. J. Theoret. Probab. 9 613-630.
  • [29] WALSH, J. B. (1986). An introduction to stochastic partial differential equations. École d'été de probabilités de St. Flour XIV. Lecture Notes in Math. 1180 265-437. Springer, New York.