Browse > Article
http://dx.doi.org/10.4134/BKMS.2010.47.1.131

INTEGRATION WITH RESPECT TO ANALOGUE OF WIENER MEASURE OVER PATHS IN WIENER SPACE AND ITS APPLICATIONS  

Ryu, Kun-Sik (Department of Mathematics Education, Han Nam University)
Publication Information
Bulletin of the Korean Mathematical Society / v.47, no.1, 2010 , pp. 131-149 More about this Journal
Abstract
In 1992, the author introduced the definition and the properties of Wiener measure over paths in Wiener space and this measure was investigated extensively by some mathematicians. In 2002, the author and Dr. Im presented an article for analogue of Wiener measure and its applications which is the generalized theory of Wiener measure theory. In this note, we will derive the analogue of Wiener measure over paths in Wiener space and establish two integration formulae, one is similar to the Wiener integration formula and another is similar to simple formula for conditional Wiener integral. Furthermore, we will give some examples for our formulae.
Keywords
analogue of Wiener measure; measure-valued measure; Bartle integral; Bochner integral; stochastically independent; conditional expectation;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
1 J. Kuelbs and R. Lepage, The law of the iterated logarithm for Brownian motion in a Banach space, Trans. Amer. Math. Soc. 185 (1973), 253–265.   DOI   ScienceOn
2 H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, Vol. 463. Springer-Verlag, Berlin-New York, 1975.
3 K. S. Ryu and M. K. Im, The measure-valued Dyson series and its stability theorem, J. Korean Math. Soc. 43 (2006), no. 3, 461–489.   DOI
4 K. S. Ryu and S. H. Shim, The rotation theorem on analogue of Wiener space, Honam Math. J. 29 (2007), no. 4, 577–588.   DOI   ScienceOn
5 K. S. Ryu and S. C. Yoo, The existence theorem and formula for an operator-valued function space integral over paths in abstract Wiener space, Houston J. Math. 28 (2002), no. 3, 599–620.
6 N. Wiener, Differential space, J. Math. Phys. 2 (1923), 131–174.   DOI
7 J. Yeh, Stochastic Processes and the Wiener Integral, Pure and Applied Mathematics, Vol. 13. Marcel Dekker, Inc., New York, 1973.
8 R. G. Laha and V. K. Rohatgi, Probability Theory, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-Chichester-Brisbane, 1979.
9 D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3795–3811.   DOI   ScienceOn
10 J. Diestel and J. J. Uhl Jr., Vector Measures, With a foreword by B. J. Pettis. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
11 M. Loeve, Probability Theory. Foundations. Random sequences, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955.
12 C. Park and D. L. Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), no. 2, 381–394.   DOI
13 K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London 1967.
14 W. J. Padgett and R. L. Taylor, Laws of Large Numbers for Normed Linear Spaces and Certain Frechet Spaces, Lecture Notes in Mathematics, Vol. 360. Springer-Verlag, Berlin-New York, 1973.
15 K. S. Ryu, The Wiener integral over paths in abstract Wiener space, J. Korean Math. Soc. 29 (1992), no. 2, 317–331.
16 K. S. Ryu and M. K. Im, A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula, Trans. Amer. Math. Soc. 354 (2002), no. 12, 4921–4951.   DOI   ScienceOn
17 M. K. Im and K. S. Ryu, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (2002), no. 5, 801–819.   DOI   ScienceOn
18 X. Fernique, Int´egrabilit´e des vecteurs gaussiens, C. R. Acad. Sci. Paris Ser. A-B 270 (1970), A1698–A1699.
19 L. Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1 pp. 31–42 Univ. California Press, Berkeley, Calif., 1967.