Lévy area of Wiener processes in Banach spaces
M. Ledoux, T. Lyons and Z. Qian
Source: Ann. Probab. Volume 30, Number 2 (2002), 546-578.
Abstract
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.
Primary Subjects: 60H10
Secondary Subjects: 60H15, 60J60, 60G15
Keywords: Brownian motion; differential equation; Gaussian comparison theorem; Gaussian measure; rough path
Full-text: Access granted (open access)
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1023481002
Mathematical Reviews number (MathSciNet):
MR1905851
Digital Object Identifier: doi:10.1214/aop/1023481002
Zentralblatt MATH identifier:
1016.60071
References
[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.
Mathematical Reviews (MathSciNet):
MR1488138
[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.
Zentralblatt MATH:
0979.60044
[7] CHEVET, S. (1977). Un résultat sur les mesures gaussiennes. C. R. Acad. Sci. Paris Sér. I Math. 284 441-444.
Zentralblatt MATH:
0362.28007
[8] COUTIN, L. and QIAN,(2002). Stochastic analysis, rough path analysis and fractional Brownian motion. Probab. Theory Related Fields 122 108-140.
Mathematical Reviews (MathSciNet):
MR1883719
[9] DA PRATO, G. and ZABCZYK, J. (1992). Stochastic equations in infinite dimensions. In Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet):
MR95g:60073
[10] DUDLEY, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 290-330.
Mathematical Reviews (MathSciNet):
MR36:3405
[11] FERNIQUE, X. (1970). Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris Sér. I Math. 270 1698-1699.
Mathematical Reviews (MathSciNet):
MR42:1170
[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.
Zentralblatt MATH:
0695.60040
[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.
Mathematical Reviews (MathSciNet):
MR99f:60145
Zentralblatt MATH:
0936.60073
[17] KUO, H.-H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Math. 436. Springer, New York.
Mathematical Reviews (MathSciNet):
MR57:1628
[18] LANDAU, H. J. and SHEPP, L. A. (1970). On the supremum of a Gaussian process. Sankhya Ser. A 32 369-378.
Mathematical Reviews (MathSciNet):
MR44:3381
Zentralblatt MATH:
0218.60039
[19] LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Springer, New York.
Mathematical Reviews (MathSciNet):
MR93c:60001
[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.
Zentralblatt MATH:
0983.60026
[22] LIFSHITS, M. A. (1995). Gaussian Random Functions. Kluwer, Dordrecht.
Mathematical Reviews (MathSciNet):
MR98k:60059
[23] LYONS, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215-310.
Mathematical Reviews (MathSciNet):
MR2000c:60089
[24] LYONS, T. J. and QIAN,(1998). Flows of diffeomorphisms induced by a geometric multiplicative functionals. Probab. Theory Related Fields 112 91-119.
Mathematical Reviews (MathSciNet):
MR99k:60153
[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.
Zentralblatt MATH:
0776.60101
[28] STOLZ, W. (1996). Some small ball probabilities for Gaussian processes under non-uniform norms. J. Theoret. Probab. 9 613-630.
Mathematical Reviews (MathSciNet):
MR97h:60036
[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.
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