Operator-valued Feynman integral via conditional Feynman integrals on a function space

Let C ₀ ^r 0; t] denote the analogue of the r-dimensional Wiener space, define X _t : C ^r 0; t] → ?^2r by X _t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X _t . Using this formula, we evaluate the conditional analytic Feynman integral for the functional $$ \Gamma _t \left( x \right) = exp \left\{ {\int_0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\}\varphi \left( {x\left( t \right)} \right) x \in C^r \left {0,t} \right] $$ , where η is a complex Borel measure on 0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ? ^r . We then introduce an integral transform as an analytic operator-valued Feynman integral over C ^r 0, t], and evaluate the integral transform for the function Γ _t via the conditional analytic Feynman integral as a kernel.