A formula for $E_ W \textrm {exp}(-2^ {-1}a^ 2\Vert x+y\Vert ^ 2_ 2)$
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- by Tzuu-Shuh Chiang, Yun Shyong Chow and Yuh-Jia Lee PDF
- Proc. Amer. Math. Soc. 100 (1987), 721-724 Request permission
Abstract:
We prove that for a complex number $a$ with $\operatorname {Re} {a^2} > - {\pi ^2}/4$ and $x( \cdot ) \in {L^2}[0,1]$, \[ {E_W}\{ \exp ( - {2^{ - 1}}{a^2}||x + y||_2^2)\} = {(\cosh a)^{ - 1/2}}\exp \left [ {{2^{ - 1}}\left ( {\int _0^1 {\int _0^1 {k(s,t)x(s)x(t)dsdt} - {a^2}\int _0^1 {{x^2}(t)dt} } } \right )} \right ],\], where $W$, the standard Wiener measure on $C[0,1]$, is the distribution of $y$ and \[ k(s,t) = {a^3}{(2\cosh a)^{ - 1}}[\sinh (a(1 - |s - t|)) - \sinh (a(1 - |s + t|))].\].References
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Additional Information
- © Copyright 1987 American Mathematical Society
- 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