Applications of the Fourier-Wiener transform to differential equations on infinite-dimensional spaces. I

Abstract:

Let $(H,i,B)$ be an abstract Wiener space and ${p_t}$ be the Wiener measure on B with variance t. Let [B] be the complexification of B and ${\mathcal {E}_a}$ be the class of exponential type analytic functions defined on [B]. We define the Fourier-Wiener c-transform for any f in ${\mathcal {E}_a}$ by \[ {F_c}f(y) = \int _\textbf {B} {f(x + iy){p_c}(dx)} \] and the inverse transform by $\mathcal {F}_c^{ - 1}f(y) = {\mathcal {F}_c}f( - y)$. Then the inversion formula holds and ${\mathcal {F}_2}$ extends to ${L^2}(B,{p_1})$ as a unitary operator. Next, we apply the above transform to investigate the existence, uniqueness and regularity of solutions for Cauchy problems associated with the following two equations: (1) ${u_t} = - {\mathcal {N}^k}u$, (2) ${u_{tt}} = - {\mathcal {N}^k}u$; and the elliptic type equation (3) $- {N^k}u = f(k \geqslant 1)$, where $\Delta$ is the Laplacian and $\mathcal {N}u(x) = - \Delta u(x) + (x,Du(x))$.