Summary
In this paper, we observe how Lévy's stochastic area looks when we see it through various topologies in the Wiener space. Our theorem implies that it is quite natural from the viewpoint of topology to define a distinct skeleton of Lévy's stochastic areaS(w) for each distinct topology in the Wiener space, or equivalently, for each distinct abstract Wiener space on which the Wiener measure andS(w) are realized. Thus we cannot determine its intrinsic skeleton in the theory of abstract Wiener spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Borell, C.: Convex measures on locally convex spaces. Ark. Mat.12, (2) 239–252 (1974)
Gross, L.: Measurable functions on Hilbert space. Trans. Am. Math. Soc.105, 372–390 (1962)
Gross, L.: Abstract Wiener spaces. Proceedings of the Fifth Berkeley Symposium on Math. Statist. and Prob. II, Part I. University of California pp. 31–42, 1967
Das Gupta, S., Eaton, M.L., Olkin, I., Perlman, M., Savage, L.J., Sobell, M.: Inequalities on the probability content of convex regions for elliptically contoured distributions, Proc. of Sixth Berkeley Symp. on Math. Statist. and Prob. II. University of California, pp. 241–264, 1972
Hida, T.: Brownian motion. New York Heidelberg Berlin. Springer 1980
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, 2nd edn. Amsterdam Tokyo: North-Holland Kodansha 1989
Itô, K., Nisio, M.: On the convergence of sums of independent Banach space valued random variables. Osaka J. Math.5, 35–48 (1968)
Kallianpur, G.: Some remarks on Hu and Meyer's paper and finitely additive canonical Hilbert space. (preprint)
Pit, L.D.: A Gaussian correlation inequality for symmetric convex sets. Ann. Probab.5, 470–474 (1977)
Shepp, L.A., Zeitouni, O.: A note on conditional exponential moments and Onsager Machlup functionals. (preprint)
Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle, Proc. of the Sixth Berkeley Symp. on Math. Statist. and Prob. III. University of California 333–359, 1972
Sugita, H.: Hu-Meyer's multiple Stratonovich integral and essential continuity of multiple Wiener integral. Bull. Sci. Math., II. Sér.113, 463–474 (1989)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sugita, H. Various topologies in the Wiener space and Lévy's stochastic area. Probab. Th. Rel. Fields 91, 283–296 (1992). https://doi.org/10.1007/BF01192058
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192058