Journal of the
Korean Mathematical Society JKMS

The Cameron--Martin translation theorem describes how Wiener measure changes under translation by elements of the Cameron--Martin space in an abstract Wiener space (AWS). Translation theorems for the analytic Feynman integrals also have been established in the literature. In this article, we derive a more general translation theorem for the analytic Feynman integral associated with bounded linear operators (B.L.OP.) on AWSs. To do this, we use a certain behavior which exists between the analytic Fourier--Feynman transform (FFT) and the convolution product (CP) of functionals on AWS. As an interesting application, we apply this translation theorem to evaluate the analytic Feynman integral of the functional \[ F(x)=\exp\bigg(-iq \int_0^Tx(t)y(t)d t \bigg),\quad y\in C_0[0,T], \, q\in\mathbb R\setminus\{0\} \] defined on the classical Wiener space $C_0[0,T]$.

Keywords: Abstract Wiener space,
Cameron--Martin translation theorem,
bounded linear operator,
analytic Feynman integral,
analytic Fourier--Feynman transform,
convolution product.

MSC numbers: Primary 46B09, 46G12; Secondary 28C20, 44A15, 42B10