International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 4, Pages 231-237
doi:10.1155/S0161171201004537
A change of scale formula for Wiener integrals of cylinder functions on the abstract Wiener space II
BK-21 Mathematical Science Division, Department of Mathematics, Seoul National University, Seoul 151-742, South Korea
Received 24 January 2000
Copyright © 2001 Young Sik Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We show that for certain bounded cylinder functions of the form F(x)=μˆ((h1,x)∼,...,(hn,x)∼), x∈B where μˆ:ℝn→ℂ is
the Fourier-transform of the complex-valued Borel measure μ on ℬ(ℝn), the Borel σ-algebra of
ℝn with ‖μ‖<∞, the analytic Feynman
integral of F exists, although the analytic Feynman integral, limz→−iqIaw(F;z)=limz→−iq(z/2π)n/2∫ℝnf(u→)exp{−(z/2)|u→|2}du→, do not
always exist for bounded cylinder functions F(x)=f((h1,x)∼,...,(hn,x)∼), x∈B. We prove a change of scale formula for Wiener integrals of F on
the abstract Wiener space.