A formula for 𝐸_{𝑊}𝑒𝑥𝑝(-2⁻¹𝑎²‖𝑥+𝑦‖²₂)

Abstract

We prove that for a complex number a a with Re ⁡ a 2 > − π 2 / 4 \operatorname {Re} {a^2} > - {\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 \[ k ( s , t ) = a 3 ( 2 cosh ⁡ a ) − 1 [ sinh ⁡ ( a ( 1 − | s − t | ) ) − sinh ⁡ ( a ( 1 − | s + t | ) ) ] . k(s,t) = {a^3}{(2\cosh a)^{ - 1}}[\sinh (a(1 - |s - t|)) - \sinh (a(1 - |s + t|))]. \] .