Notes on analytic Feynman integrable functionals. (English) Zbl 0808.28009
The authors prove the Feynman integrability for a wide class of functionals defined on the multiparameter Wiener space.
Let \(m^ \nu_ N\) be the Wiener measure on the space of functions on \([0,T]^ N\) that are continuous and take values in the \(\nu\)-dimensional space \(R^ \nu\). Let further \(F: C^ \nu_ N\to C\) be a functional such that for each \(\lambda> 0\) the Wiener integral \[ J(\lambda)= \int_{C^ \nu_ N} F(\lambda^{-1/2} x) dm^ \nu_ N(x) \] exists. If there exists an analytic continuation \(J^*\) of \(J\) onto the right half-plane \(C^ +\), then \(J^*\) is called the analytic Wiener integral. If now for a real non-zero \(q\) there exists \(\lim_{\lambda\to -iq} J^*(\lambda)\), then this limit is called the analytic Feynman integral of \(F\) with parameter \(q\). There are given conditions ensuring the analytic Feynman integrability for a wide class of functionals. In particular, the authors generalize a result by the second author, G. W. Johnson and D. L. Skoug [Pac. J. Math. 122, 11-33 (1986; Zbl 0594.28013)].
Let \(m^ \nu_ N\) be the Wiener measure on the space of functions on \([0,T]^ N\) that are continuous and take values in the \(\nu\)-dimensional space \(R^ \nu\). Let further \(F: C^ \nu_ N\to C\) be a functional such that for each \(\lambda> 0\) the Wiener integral \[ J(\lambda)= \int_{C^ \nu_ N} F(\lambda^{-1/2} x) dm^ \nu_ N(x) \] exists. If there exists an analytic continuation \(J^*\) of \(J\) onto the right half-plane \(C^ +\), then \(J^*\) is called the analytic Wiener integral. If now for a real non-zero \(q\) there exists \(\lim_{\lambda\to -iq} J^*(\lambda)\), then this limit is called the analytic Feynman integral of \(F\) with parameter \(q\). There are given conditions ensuring the analytic Feynman integrability for a wide class of functionals. In particular, the authors generalize a result by the second author, G. W. Johnson and D. L. Skoug [Pac. J. Math. 122, 11-33 (1986; Zbl 0594.28013)].
Reviewer: S.A.Chobanjan (Tbilisi)
MSC:
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
Citations:
Zbl 0594.28013References:
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