An operator-valued Yeh-Wiener integral and a Kac-Feynman Wiener integral equation
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- by Chull Park and David Skoug PDF
- Proc. Amer. Math. Soc. 120 (1994), 929-942 Request permission
Abstract:
Let $C[0,T]$ denote Wiener space, i.e., the space of all continuous functions $\eta (t)$ on $[0,T]$ such that $\eta (0) = 0$. For $Q = [0,S] \times [0,T]$ let $C(Q)$ denote Yeh-Wiener space, i.e., the space of all $\mathbb {R}$-valued continuous functions $x(s,t)$ on $Q$ such that $x(0,t) = x(s,0) = 0$ for all $(s,t)$ in $Q$. For $h \in {L_2}(Q)$ let $Z(x;s,t)$ be the Gaussian process defined by the stochastic integral \[ Z(x;s,t) = \int _0^t {\int _0^s {h(u,v)dx(u,v).} } \] Then for appropriate functionals $F$ and $\psi$ we show that the operator-valued function space integral \[ (I_\lambda ^h(F)\psi )(\eta ( \cdot )) = {E_x}[F({\lambda ^{ - 1/2}}Z(x; \cdot , \cdot ) + \eta ( \cdot ))\psi ({\lambda ^{ - 1/2}}Z(x;S, \cdot ) + \eta ( \cdot ))]\] is the unique solution of a Kac-Feynman Wiener integral equation. We also use this integral equation to evaluate various Yeh-Wiener integrals.References
- R. H. Cameron and D. A. Storvick, An operator valued function space integral and a related integral equation, J. Math. Mech. 18 (1968), 517–552. MR 0236347, DOI 10.1512/iumj.1969.18.18041
- R. H. Cameron and D. A. Storvick, An integral equation related to the Schroedinger equation with an application to integration in function space, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 175–193. MR 0348071 —, An operator valued function space integral applied to functions of class ${L_2}$, J. Math. Anal. Appl. 42 (1973), 330-373.
- R. H. Cameron and D. A. Storvick, An operator valued Yeh-Wiener integral, and a Wiener integral equation, Indiana Univ. Math. J. 25 (1976), no. 3, 235–258. MR 399403, DOI 10.1512/iumj.1976.25.25020
- Dong Myung Chung, Chull Park, and David Skoug, Operator-valued Feynman integrals via conditional Feynman integrals, Pacific J. Math. 146 (1990), no. 1, 21–42. MR 1073518
- J. L. Doob, Stochastic processes, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1953 original; A Wiley-Interscience Publication. MR 1038526
- Takeyuki Hida, Brownian motion, Applications of Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1980. Translated from the Japanese by the author and T. P. Speed. MR 562914
- G. W. Johnson and D. L. Skoug, A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger’s equation, J. Functional Analysis 12 (1973), 129–152. MR 0348072, DOI 10.1016/0022-1236(73)90019-0
- Chull Park, On Fredholm transformations in Yeh-Wiener space, Pacific J. Math. 40 (1972), 173–195. MR 304604
- Chull Park and David Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), no. 2, 381–394. MR 968620
- Chull Park and David Skoug, Conditional Yeh-Wiener integrals with vector-valued conditioning functions, Proc. Amer. Math. Soc. 105 (1989), no. 2, 450–461. MR 960650, DOI 10.1090/S0002-9939-1989-0960650-6
- J. Yeh, Wiener measure in a space of functions of two variables, Trans. Amer. Math. Soc. 95 (1960), 433–450. MR 125433, DOI 10.1090/S0002-9947-1960-0125433-1 —, Stochastic processes and the Wiener integral, Marcel Dekker, New York, 1983.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 929-942
- MSC: Primary 28C20; Secondary 47N30, 60J65
- DOI: https://doi.org/10.1090/S0002-9939-1994-1213867-1
- MathSciNet review: 1213867