|
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
MathSciNet review:
1926843
Retrieve article in:
PDF
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
|
