Given two Wiener measurable
functionals X and Y on the Wiener space C[0,t], of which the latter is Wiener
integrable, the conditional Wiener integral of Y given X is defined as the conditional
expectation Ew(Y |X) given as a function on the value space of X. Several Fourier
inversion formulae for retrieving the conditional Wiener integral Ew(Y |X) in which
X[x] = x(t) for x ∈ C[0,t] are derived. Examples of evaluation of Ew(Y |X) are given.
It is shown that the Kac-Feynman formula can be derived by applying an inversion
formula to Ew(Y |X) where
![∫
t
Y[x] = exp{− 0 V [x(s)]ds}.](a330x.png)
