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We prove that for a complex number      a a       with      Re ⁡  a 2 &gt; −  π  2 / 4 \operatorname {Re} {a^2} &gt; - {\pi ^2}/4       and      x ( ⋅  ) ∈  L 2 [ 0 , 1 ] x( \cdot ) \in {L^2}[0,1]      ,  \[    E W { exp ⁡  ( −  2 −  1 a 2 | | x + y | | 2 2 ) } = ( cosh ⁡  a ) −  1 / 2 exp ⁡  [ 2 −  1 ( ∫  0 1 ∫  0 1 k ( s , t ) x ( s ) x ( t ) d s d t −  a 2 ∫  0 1 x 2 ( t ) d t ) ] , {E_W}\{ \exp ( - {2^{ - 1}}{a^2}||x + y||_2^2)\} = {(\cosh a)^{ - 1/2}}\exp \left [ {{2^{ - 1}}\left ( {\int _0^1 {\int _0^1 {k(s,t)x(s)x(t)dsdt} - {a^2}\int _0^1 {{x^2}(t)dt} } } \right )} \right ],     \] , where      W W      , the standard Wiener measure on      C [ 0 , 1 ] C[0,1]      , is the distribution of      y y       and  \[    k ( s , t ) = a 3 ( 2 cosh ⁡  a ) −  1 [ sinh ⁡  ( a ( 1 −  | s −  t | ) ) −  sinh ⁡  ( a ( 1 −  | s + t | ) ) ] . k(s,t) = {a^3}{(2\cosh a)^{ - 1}}[\sinh (a(1 - |s - t|)) - \sinh (a(1 - |s + t|))].     \] .

A formula for 𝐸_{𝑊}𝑒𝑥𝑝(-2⁻¹𝑎²‖𝑥+𝑦‖²₂)