Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-25T15:17:11.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Sequential Gaussian Markov Integrals*

Published online by Cambridge University Press:  22 January 2016

John A. Beekman
John A. Beekman*
Affiliation:
Ball State University, Muncie, Indiana
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [6] R.H. Cameron defined and studied a sequential Wiener integral. This was motivated by the function space integral R.P. Feynman used in [12] to give a solution to the Schröedinger equation. In [5] the present author studied sequential Gaussian Markov integrals with a positive parameter. This paper gives sufficient conditions on the integrand for such integrals to exist, when the parameter is complex. These sequential integrals are related to ordianry Gaussian Markov integrals through a Fourier transform type formula extended from [5]. We shall show that such integrals are equal to conditional Wiener integrals of suitably modified functionals.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 42 , March 1971 , pp. 9 - 21
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

Footnotes

*

This research was partially supported by the National Science Foundation through grants NSF GP-7639 and NSF GP-9634.

References

[1] Beekman, John A., “Solutions to Generalized Schroedinger Equations via Feynman Integrals Connected with Gaussian Markov Stochastic Processes,” Thesis, Univ. of Minnesota, 1963.Google Scholar
[2] Beekman, John A., “Gaussian Processes and Generalized Schroedinger Equations,” J. Math. Mech. 14 (1965), 789–806.Google Scholar
[3] Beekman, John A., “Gaussian Markov Processes and a Boundary Value Problem,” Trans. Amer. Math. Soc. 126 (1967), 29–42.Google Scholar
[4] Beekman, John A., “Feynman-Cameron Integrals,” J. Math, and Phys. 46 (1967), 253–266.Google Scholar
[5] Beekman, John A., “Green’s Functions for Generalized Schroedinger Equations,” Nagoya Math. J. 35 (1969), 133–150. Correction: Nagoya Math. J. 39 (1970), 199.Google Scholar
[6] Cameron, R.H., “A Family of Integrals Serving to Connect the Wiener and Feynman Integrals,” J. Math, and Phys. 39 (1960), 126–140.Google Scholar
[7] Cameron, R.H., “The Ilstow and Feynman Integrals,” J. Analyse Math. 10 (1962–3), 287–361.Google Scholar
[8] Cameron, R.H., “Approximations to Certain Feynman Integrals,” J. Analyse Math. 21 (1968), 337–371.Google Scholar
[9] Cameron, R.H. and Storvick, D.A., “An Operator Valued Function Space Integral and a Related Integral Equation,” J. Math. Mech. 18 (1968), 517–552.Google Scholar
[10] Copson, E.T., Theory of Functions of a Complex Variable, Oxford, 1935.Google Scholar
[11] Darling, D.A. and Siegert, A.J.F., “Integral Equations for the Characteristic Functions of Certain Functionals of Multi-dimensional Markoff Processes,” Rand Report, P-429, Rand Corporation, Santa Monica, California, 1955.Google Scholar
[12] Feynman, R.P., “Space-Time Approach to Non-relativistic Quantum Mechanics,” Rev. Modern Phys. 20 (1948), 367–387.Google Scholar
[13] Fosdick, L.D., “Numerical Estimation of the Partition Function in Quantum Statistics,” J. Mathematical Phys. 3 (1962), 1251-1264.Google Scholar
[14] Fosdick, L.D., “Approximation of a Class of Wiener Integrals,” Math. Comp. 19 (1965), 225–233.Google Scholar
[15] Fosdick, L.D. and Jordan, H.F., “Approximation of a Conditional Wiener Integral,” J. Computational Phys. 3 (1968), 1–16.CrossRefGoogle Scholar
[16] Peña-Auerbach, L. de la and Garcia-Colin, L.S., “Simple Generalization of Schrödinger’s Equations,” J. Mathematical Phys. 9 (1968), 922–927.Google Scholar
[17] Tsuda, Takao, Ichida, Kozo, and Kiyono, Takeshi, “Monte Carlo Path-Integral Calculations for Two-Point Boundary-Value Problems,” Numer. Math. 10 (1967), 110–116.Google Scholar
[18] Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis, Cambridge, 1927.Google Scholar