Integral transforms of analytic functions on abstract Wiener spaces

Let (H, B) be an abstract Wiener pair and p_t the Wiener measure with variance t. Let $\text{E}$_a be the class of exponential type analytic functions defined on the complexification B] of B. For each pair of nonzero complex numbers α, β and f ? $\text{E}$_a, we define

$\text{F}{}_{\mathrm{\alpha ,\beta }}\text{f(y)=}\text{∫}\text{B}\text{f(αx+βy)p}{}_1\text{(dx) (y ?B]).}$

We show that the inverse $\text{F}$_α,β^?1 exists and there exist two nonzero complex numbers α′,β′ such that

. Clearly, the Fourier-Wiener transform, the Fourier-Feynman transform, and the Gauss transform are special cases of $\text{F}$_α,β. Finally, we apply the transform to investigate the existence of solutions for the differential equations associated with the operator $\text{N}$_c, where c is a nonzero complex number and $\text{N}$_c is defined by

$\text{N}{}_{\mathrm{c}}\text{u(x)=?Δu(x)+c(x,Du(x))}$

where Δ is the Laplacian and (·, ·) is the $\text{B-B}{}^1$ pairing. We show that the solutions can be represented as integrals with respect to the Wiener measure.