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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An operator-valued Yeh-Wiener integral and a Kac-Feynman Wiener integral equation


Authors: Chull Park and David Skoug
Journal: Proc. Amer. Math. Soc. 120 (1994), 929-942
MSC: Primary 28C20; Secondary 47N30, 60J65
MathSciNet review: 1213867
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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

$\displaystyle 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

$\displaystyle (I_\lambda ^h(F)\psi )(\eta ( \cdot )) = {E_x}[F({\lambda ^{ - 1/... ...t ) + \eta ( \cdot ))\psi ({\lambda ^{ - 1/2}}Z(x;S, \cdot ) + \eta ( \cdot ))]$

is the unique solution of a Kac-Feynman Wiener integral equation. We also use this integral equation to evaluate various Yeh-Wiener integrals.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1213867-1
PII: S 0002-9939(1994)1213867-1
Keywords: Yeh-Wiener itegral, Wiener integral equation, Gaussian process
Article copyright: © Copyright 1994 American Mathematical Society