Translation Theorem for Conditional Function Space Integrals and Applications

The conditional Feynman integral provides solutions to integral equations equivalent to heat and Schrödinger equations. The Cameron–Martin translation theorem illustrates how the Wiener measure changes under translation via Cameron–Martin space elements in abstract Wiener space. Translation theorems for analytic Feynman integrals have been established in many research articles. This study aims to present a translation theorem for the conditional function space integral of functionals on the generalized Wiener space $C_{a,b}[0,T]$ induced via a generalized Brownian motion process determined using continuous functions $a\left(t\right)$ and $b\left(t\right)$. As an application, we establish a translation theorem for the conditional generalized analytic Feynman integral of functionals on $C_{a,b}[0,T]$. We then provide explicit examples of functionals on $C_{a,b}[0,T]$ to which the conditional translation theorem on $C_{a,b}[0,T]$ can be applied. Our formulas and results are more complicated than the corresponding formulas and results in the previous research on the Wiener space $C_0[0,T]$ because the generalized Brownian motion process used in this study is neither stationary in time nor centered. In this study, the stochastic process used is subject to a drift function.