|
A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula
Author(s):
K.
S.
Ryu;
M.
K.
Im
Journal:
Trans. Amer. Math. Soc.
354
(2002),
4921-4951.
MSC (2000):
Primary 28C35, 28C20, 45D05, 47A56
Posted:
July 23, 2002
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this article, we consider a complex-valued and a measure-valued measure on ![$C [0,t]$](/tran/2002-354-12/S0002-9947-02-03077-5/gif-abstract0/img1.gif) , the space of all real-valued continuous functions on ![$[0,t]$](/tran/2002-354-12/S0002-9947-02-03077-5/gif-abstract0/img2.gif) . Using these concepts, we establish the measure-valued Feynman-Kac formula and we prove that this formula satisfies a Volterra integral equation. The work here is patterned to some extent on earlier works by Kluvanek in 1983 and by Lapidus in 1987, but the present setting requires a number of new concepts and results.
References:
-
- 1.
- Burrill, C. W., Measure, integration and probability, McGraw-Hill, New York, 1972. MR 56:15862
- 2.
- Cameron, R. H. and Storvick, D. A., An operator-valued function space integral and a related integral equation, J. Math. Mech. 18, 1968, 517-552. MR 38:4643
- 3.
- Cohn, D. L., Measure theory, Birkhäuser, Boston, 1980. MR 81k:28001
- 4.
- Diestel, J. and Uhl, J. J., Vector measures, Mathematical Survey, Amer. Math. Soc., 1977. MR 56:12216
- 5.
- Dunford, N. and Schwartz, J. T., Linear Operators, part I, General Theory, Pure and Applied Mathematics, Vol. VII, Wiley Interscience, New York, 1958. MR 22:8302
- 6.
- Halmos, P. R., Measure Theory, Springer-Verlag, New York, 1950. MR 11:504d
- 7.
- Hewitt, E. and Stromberg, K., Real and Abstract Analysis, Springer-Verlag, New York, 1965. MR 32:5826
- 8.
- Johnson, G. W. and Lapidus, M. L., Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman's operational calculus, Memoirs Amer. Math. Soc., 62, No. 351, 1986, 1-78. MR 88f:81034
- 9.
- Johnson, G. W. and Lapidus, M. L., The Feynman integral and Feynman's operational calculus, Oxford Mathematical Monographs, Oxford Univ. Press, 2000. MR 2001i:58015
- 10.
- Kluvanek, I., Operator valued measures and perturbations of semi-groups, Arch. Rational Mech. Anal. 81-82, 1983, 161-180. MR 84j:28019
- 11.
- Kluvanek, I. and Knowles, G., Vector measures and control systems, Math. Studies, No 20, Amsterdam, North-Holland, 1975. MR 58:17033
- 12.
- Lapidus, M. L., The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus, Stud. Appl. Math., 76, 1987, 93-132. MR 89j:81057
- 13.
- Lapidus, M. L., Strong product integration of measures and the Feynman-Kac formula with a Lebesgue-Stieltjes measure, Circ. Math. Palermo (2) Suppl. 17 (1987), 271-312. MR 90c:28021
- 14.
- Lapidus, M. L., The Feynman-Kac formula with a Lebesgue-Stieltjes measure: An integral equation in the general case, Integral Equations and Operator Theory, 12 (1989), 163-210. MR 92e:47064
- 15.
- Lewis, D. R., Integration with respect to vector measure, Pacific J. Math., 33, No 1, 1970, 157-165. MR 41:3706
- 16.
- Novinger, W. P., Mean convergence in
 space, Proc. Amer. Math. Soc., 34, 1972, 627-628. MR 45:3665 - 17.
- Okikiolu, G. G., Aspect of the theory of bounded linear operators in
 space, Academic Press, London, 1971. MR 56:3581 - 18.
- Parthasarathy, K. R., Probability measures on metric spaces, Academic Press, New York, 1967. MR 37:2271
- 19.
- Rudin, W., Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987. MR 88k:00002
- 20.
- Wiener, N., Differential space, J. Math. Phys., 2, 1923, 131-174.
- 21.
- Yeh, J., Inversion of conditional expectations, Pacific J. Math., 52, no. 2, 1974, 631-640. MR 51:1896
- 22.
- Yeh, J., Stochastic processes and the Wiener integral, Marcel Deckker, New York, 1973. MR 57:14166
- 23.
- Yosida, K., Functional Analysis, 4th Edition, Springer-Verlag Berlin, 1974. MR 50:2851
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
28C35, 28C20, 45D05, 47A56
Retrieve articles in all Journals with MSC
(2000):
28C35, 28C20, 45D05, 47A56
Additional Information:
K.
S.
Ryu
Affiliation:
Department of Mathematics, Han Nam University, Taejon 306-791, Korea
Email:
ksr@math.hannam.ac.kr
M.
K.
Im
Affiliation:
Department of Mathematics, Han Nam University, Taejon 306-791, Korea
Email:
mki@mail.hannam.ac.kr
DOI:
10.1090/S0002-9947-02-03077-5
PII:
S 0002-9947(02)03077-5
Keywords:
Analogue of Wiener measure,
Bartle integral,
measure-valued Feynman-Kac formula,
Volterra integral equation
Received by editor(s):
December 18, 2001
Received by editor(s) in revised form:
April 1, 2002
Posted:
July 23, 2002
Dedicated:
Dedicated to Professor Kun Soo Chang on his sixtieth birthday
Copyright of article:
Copyright
2002,
American Mathematical Society
Forward Citation(s): Information for authors on submitting citations The following works have cited this article K.S.Ryu and M.K.Im, An analogue of Wiener measure and its applications, Journal of Korean Mathematical Society Vol. 35,No.5 (2002), 801 - 819.
|
Comments: webmaster@ams.org