Journal of the Korean Mathematical Society (대한수학회지)

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TRANSLATION THEOREM FOR THE ANALYTIC FEYNMAN INTEGRAL ASSOCIATED WITH BOUNDED LINEAR OPERATORS ON ABSTRACT WIENER SPACES AND AN APPLICATION

  • Jae Gil Choi (Department of Mathematics Basic Sciences and Mathematics Center Dankook University)
  • Received : 2023.08.14
  • Accepted : 2024.05.08
  • Published : 2024.09.01
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Abstract

The Cameron-Martin translation theorem describes how Wiener measure changes under translation by elements of the Cameron-Martin space in an abstract Wiener space (AWS). Translation theorems for the analytic Feynman integrals also have been established in the literature. In this article, we derive a more general translation theorem for the analytic Feynman integral associated with bounded linear operators (B.L.OP.) on AWSs. To do this, we use a certain behavior which exists between the analytic Fourier-Feynman transform (FFT) and the convolution product (CP) of functionals on AWS. As an interesting application, we apply this translation theorem to evaluate the analytic Feynman integral of the functional $$F(x)={\exp}\left(-iq\int_{0}^{T}x(t)y(t)dt\right),\,y{\in}C_0[0,\,T],\;q{\in}{\mathbb{R}}\,{\backslash}\,\{0\}$$ defined on the classical Wiener space C0[0, T].

Keywords

Acknowledgement

The author would like to express his gratitude to the editor and the referees for their valuable comments and suggestions which have improved the original paper.

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