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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Yuh Jia Lee
Journal: Trans. Amer. Math. Soc. 262 (1980), 259-283
MSC: Primary 35R15; Secondary 28A25, 35C15
MathSciNet review: 583855
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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

$\displaystyle {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))$.

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Keywords: Abstract Wiener space, Wiener measure, Fourier-Wiener c-transform, exponential type analytic function, equicontinuous semigroup of class ($ ({C_0})$)
Article copyright: © Copyright 1980 American Mathematical Society