Translation Theorem for Conditional Function Space Integrals and Applications

The aim of this paper is to provide a translation theorem for conditional function space integral of functionals on the generalized Wiener space Ca,b[0,T] induced by a generalized Brownian motion process. As an application, we establish a translation theorem for the conditional generalized analytic Feynman integral of functionals on Ca,b[0,T]. We then provide explicit examples of functionals on Ca,b[0,T] to which the conditional translation theorem on Ca,b[0,T] can be applied. The formulas and the results in this paper are more complicated than the corresponding formulas and the results in previous research on the Wiener space C0[0,T], because the generalized Brownian motion process used in this paper is neither stationary in time nor centered. The process used in this paper is subject to a drift function.