Journal of the
Korean Mathematical Society JKMS

In this article, we establish translation theorems for the analytic Fourier--Feynman transform of functionals in non-stationary Gaussian processes on Wiener space. We then proceed to show that these general translation theorems can be applied to two well-known classes of functionals; namely, the Banach algebra $\mathcal S$ introduced by Cameron and Storvick, and the space $\mathcal B_{\mathcal A}^{(p)}$ consisting of functionals of the form $F(x)=f(\langle{\alpha_1,x}\rangle,\ldots,\langle{\alpha_n,x}\rangle)$, where $\langle{\alpha,x}\rangle$ denotes the Paley--Wiener--Zygmund stochastic integral $\int_0^T \alpha(t)dx(t)$.

Keywords: translation theorem, Gaussian process, generalized Fourier--Feynman transform, convolution product

MSC numbers: Primary 60J65, 28C20, 60G15