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A formula for \(E_ W\,\exp (-2^{-1}a^ 2\| x+y\| ^ 2_ 2)\). (English) Zbl 0641.60006
We prove that for a complex number a with Re\(a^ 2>-\pi^ 2/4\) and \(x(\cdot)\in L^ 2[0,1]\), \[ E_ W\{\exp (-2^{-1}a^ 2\| x+y\|^ 2_ 2)\}= \] \[ =(\cosh a)^{-}\exp [2^{- 1}(\int^{1}_{0}\int^{1}_{0}k(s,t)x(s)x(t)ds dt-a^ 2\int^{1}_{0}x^ 2(t)dt)], \] where W, the standard Wiener measure on \(C[0,1]\), is the distribution of y and \[ k(s,t)=a^ 3(2 \cosh a)^{-1}[\sinh (a(1-| s-t|))-\sinh (a(1-| s+t|))]. \]

MSC:
60B11 Probability theory on linear topological spaces
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