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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A formula for $ E\sb W\,{\rm exp}(-2\sp {-1}a\sp 2\Vert x+y\Vert \sp 2\sb 2)$


Authors: Tzuu-Shuh Chiang, Yun Shyong Chow and Yuh-Jia Lee
Journal: Proc. Amer. Math. Soc. 100 (1987), 721-724
MSC: Primary 60B11; Secondary 81C35
DOI: https://doi.org/10.1090/S0002-9939-1987-0894444-5
MathSciNet review: 894444
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Abstract: We prove that for a complex number $ a$ with $ \operatorname{Re} {a^2} > - {\pi ^2}/4$ and $ x( \cdot ) \in {L^2}[0,1]$,

$\displaystyle {E_W}\{ \exp ( - {2^{ - 1}}{a^2}\vert\vert x + y\vert\vert _2^2)\... ...int_0^1 {k(s,t)x(s)x(t)dsdt} - {a^2}\int_0^1 {{x^2}(t)dt} } } \right)} \right],$

, where $ W$, the standard Wiener measure on $ C[0,1]$, is the distribution of $ y$ and

$\displaystyle k(s,t) = {a^3}{(2\cosh a)^{ - 1}}[\sinh (a(1 - \vert s - t\vert)) - \sinh (a(1 - \vert s + t\vert))].$

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0894444-5
Keywords: Wiener measure, Fourier expansion
Article copyright: © Copyright 1987 American Mathematical Society