Lévy area of Wiener processes in Banach spaces.(English)Zbl 1016.60071

The second author has recently developed a new approach to an equation $$dy = f(y)dx$$ in the Euclidean space, applicable also in the case when $$x$$ is a non-smooth function, like a trajectory of a Brownian motion [see e.g., Rev. Mat. Iberoam. 14, 215-310 (1998; Zbl 0923.34056), or the book by the second and the third author, “System control and rough paths” (Oxford, 2002)]. In the paper it is shown that this theory covers also equations driven by Banach space-valued Wiener processes, which implies that the classical Stratonovich integration theory may be extended to Wiener processes in Banach spaces.

MSC:

 60J65 Brownian motion 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G15 Gaussian processes

Keywords:

Lévy area; rough paths; Brownian motion

Zbl 0923.34056
Full Text:

References:

 [1] ADLER, R. J. (1990). An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA. · Zbl 0747.60039 [2] BASS, R. F., HAMBLY, B. M. and LYONS, T. J. (1999). Extending the Wong-Zakai theorem to reversible Markov processes. · Zbl 1010.60070 [3] BRZE ŹNIAK,(1997). On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61 245-295. · Zbl 0891.60056 [4] BRZE ŹNIAK,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. · Zbl 1040.60047 [5] BRZE ŹNIAK,and ELWORTHY, D. (1999). Stochastic differential equations on Banach manifolds. · Zbl 0965.58028 [6] CAPITAINE, M. and DONATI-MARTIN, C. (2001). The Lévy area for the free Brownian motion. J. Funct. Anal. 179 153-169. · Zbl 0979.60044 [7] CHEVET, S. (1977). Un résultat sur les mesures gaussiennes. C. R. Acad. Sci. Paris Sér. I Math. 284 441-444. · Zbl 0362.28007 [8] COUTIN, L. and QIAN,(2002). Stochastic analysis, rough path analysis and fractional Brownian motion. Probab. Theory Related Fields 122 108-140. · Zbl 1047.60029 [9] DA PRATO, G. and ZABCZYK, J. (1992). Stochastic equations in infinite dimensions. In Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press. · Zbl 0761.60052 [10] DUDLEY, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 290-330. · Zbl 0188.20502 [11] FERNIQUE, X. (1970). Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris Sér. I Math. 270 1698-1699. · Zbl 0206.19002 [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. · Zbl 0331.60025 [13] GOODMAN, V. and KUELBS, J. (1991). Rates of clustering for some Gaussian self-similar processes. Probab. Theory Related Fields 88 47-75. · Zbl 0695.60040 [14] GROSS, L. (1965). Abstract Wiener spaces. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 31-42. Univ. California Press, Berkeley. · Zbl 0187.40903 [15] GROSS, L. (1970). Lecture on Modern Analysis and Applications II. Lecture Notes in Math. 140. Springer, New York. · Zbl 0203.13002 [16] HAMBLY, B. M. and LYONS, T. J. (1998). Stochastic area for Brownian motion on Sierpinski gasket. Ann. Probab. 26 132-148. · Zbl 0936.60073 [17] KUO, H.-H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Math. 436. Springer, New York. · Zbl 0306.28010 [18] LANDAU, H. J. and SHEPP, L. A. (1970). On the supremum of a Gaussian process. Sankhya Ser. A 32 369-378. · Zbl 0218.60039 [19] LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Springer, New York. · Zbl 0748.60004 [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. · Zbl 0874.60005 [21] LI, W. and LINDE, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556-1578. · Zbl 0983.60026 [22] LIFSHITS, M. A. (1995). Gaussian Random Functions. Kluwer, Dordrecht. · Zbl 0832.60002 [23] LYONS, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215-310. · Zbl 0923.34056 [24] LYONS, T. J. and QIAN,(1998). Flows of diffeomorphisms induced by a geometric multiplicative functionals. Probab. Theory Related Fields 112 91-119. · Zbl 0918.60009 [25] LYONS, T. J. and QIAN,(2000). System Control and Rough Paths. Oxford Univ. Press. · Zbl 1029.93001 [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. · Zbl 0776.60101 [28] STOLZ, W. (1996). Some small ball probabilities for Gaussian processes under non-uniform norms. J. Theoret. Probab. 9 613-630. · Zbl 0855.60039 [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. · Zbl 0608.60060
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