Abstract
Let u(t) be the operator associated by path integration with the Feynman-Kac functional in which the time integration is performed with respect to an arbitrary Borel measure η instead of ordinary Lebesgue measurel. We show that u(t), considered as a function of time t, satisfies a Volterra-Stieltjes integral equation, denoted by (*). We refer to this result as the “Feynman-Kac formula with a Lebesgue-Stieltjes measure”. Indeed, when n=l, we recover the classical Feynman-Kac formula since (*) then yields the heat (resp., Schrödinger) equation in the diffusion (resp., quantum mechanical) case. We stress that the measure η is in general the sum of an absolutely continuous, a singular continuous and a (countably supported) discrete part. We also study various properties of (*) and of its solution. These results extend and use previous work of the author dealing with measures having finitely supported discrete part (Stud. Appl. Math.76 (1987), 93–132); they seem to be new in the diffusion (or “imaginary time”) as well as in the quantum mechanical (or “real time”) case.
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References
D.L. COHN,Measure Theory, Birkhäuser, Boston, Mass., 1980.
J.D. DEVREESE & G.J. PAPADOPOULOS (Eds.),Path Integrals and their Applications in Quantum, Statistical, and Solid State Physics, Nato Advanced Study Institutes, Ser. B, Vol. 34, Plenum Press, New York, N.Y., 1978.
J.D. DOLLARD & C.N. FRIEDMAN, Product integration of measures and applications,J. Differential Equations 31(1979), 418–464.
J.D. DOLLARD & C.N. FRIEDMAN,Product Integration with Applications to Differential Equations, Encyclopedia of Mathematics and its Applications, Vol. 10, Addison-Wesley, Reading, Mass., 1979.
T. EGUCHI, P.B. GILKEY & A.J. HANSON, Gravitation, gauge theories and differential geometry,Phys. Rep. 66(1980), 213–393.
R.P. FEYNMAN, Space-time approach to non-relativistic quantum mechanics,Rev. Modern Phys. 20(1948), 367–387.
R.P. FEYNMAN, An operator calculus having applications in quantum electrodynamics,Phys. Rev. 84(1951), 108–128.
E. GETZLER, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem,Comm. Math. Phys. 92(1983), 163–178.
J. GLIMM & A. JAFFE,Quantum Physics: A Functional Integration Point of View, Springer-Verlag, New York, N.Y., 1981.
T. HIDA,Brownian Motion, Springer-Verlag, New York, N.Y., 1980.
E. HILLE & R.S. PHILLIPS,Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ., Vol. 31, rev. ed., Providence, R.I., 1957.
G.W. JOHNSON & M.L. LAPIDUS, Feynman's operational calculus, generalized Dyson series and the Feynman integral, in “Operator Algebras and Mathematical Physics” (P.E.T. Jørgensen and P. Muhly Eds.), Contemp. Math., Vol. 62, Amer. Math. Soc., Providence, R.I., 1987, pp. 437–445.
G.W. JOHNSON & M.L. LAPIDUS, Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman's operational calculus,Mem. Amer. Math. Soc., No. 351,62(1986), 1–78.
G.W. JOHNSON & M.L. LAPIDUS, Noncommutative operations on Wiener functionals and Feynman's operational calculus, to appear inJournal of Functional Analysis. [Announced in: Une multiplication non commutative des fonctionnelles de Wiener et le calcul opérationnel de Feynman,C. R. Acad. Sci. Paris Sér I Math.304(1987), 523–526.]
M. KAC, On some connections between probability theory and differential and integral equations, in “Proc. 2nd. Berkeley Symp. Math. Stat. Prob.” (J. Neyman Ed.), Univ. of California Press, Berkeley, Calif., 1951, pp. 189–215.
M. KAC,Integration in Function Spaces and some of its Applications, Lezioni Fermiane, Scuola Normale Superiore, Pisa, 1980.
M.L. LAPIDUS, The differential equation for the Feynman-Kac formula with a Lebesgue-Stieltjes measure,Lett. Math. Phys. 11(1986), 1–13.
M.L. LAPIDUS, The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus,Stud. Appl. Math. 76(1987), 93–132.
M.L. LAPIDUS,Formules de Trotter et Calcul Opérationnel de Feynman, Thèse de Doctorat d'Etat, Mathématiques, Université Pierre et Marie Curie (Paris VI), France, June 1986. (Part III: “Calcul opérationnel de Feynman.”)
M.L. LAPIDUS, Strong product integration of measures and the Feynman-Kac formula with a Lebesgue-Stieltjes measure, inSupplemento Ai Rendiconti Del Circolo Matematico Di Palermo, Proc. Summer School on “Functional Integration with Emphasis on the Feynman Integral” held at the University of Sherbrooke, Canada, July 1986 (to appear).
M.L. LAPIDUS, Fractral drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Univ. of Georgia preprint, 1988.
M.L. LAPIDUS, An axiomatic framework for Feynman's operational calculus (in preparation).
B.B. MANDELBROT,The Fractal Geometry of Nature, rev. and enl. ed., W.H. Freeman, New York, N.Y., 1983.
M. REED & B. SIMON,Methods of Modern Mathematical Physics, Vol. I,Functional Analysis, rev. and enl. ed., Academic Press, New York, N.Y., 1980.
B. SIMON,Functional Integration and Quantum Physics, Academic Press, New York, N.Y., 1978.
M. SIRUGUEet al., Jump processes: An introduction and some applications to quantum physics, inSupplemento Ai Rendiconti Del Circolo Matematico Di Palermo, Proc. Summer School on “Functional Integration with Emphasis on the Feynman Integral” held at the University of Sherbrooke, Canada, July 1986 (to appear).
R.L. WHEEDEN & A. ZYGMUND,Measure and Integral: An Introduction to Real Analysis, Marcel Dekker, New York, N.Y., 1977.
N. WIENER,Collected Works, Vol. I, MIT Press, Cambridge, Mass., 1976.
D. WILLIAMS,Diffusions, Markov Processes and Martingales, Vol. I, John Wiley & Sons, New York, N.Y., 1979.
E. WITTEN, Physics and geometry, in “Proc. Internat. Congress Math.” Berkeley, August 1986 (A.M. Gleason Ed.), Vol. I, Amer. Math. Soc., Providence, R.I., 1987, pp. 267–303.
J. YEH,Stochastic Processes and the Wiener Integral, Marcel Dekker, New York, N.Y., 1973.
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Research partially supported by the National Science Foundation under Grant DMS 8703138. This work was also supported in part by NSF Grant 8120790 at the Mathematical Sciences Research Institute in Berkeley, U.S.A., the CNPq and the Organization of Latin American States at theInstituto de Matemática Pura E Aplicada (IMPA) in Rio de Janeiro, Brazil, as well as theUniversité Pierre et Marie Curie (Paris VI) and the Université Paris Dauphine in Paris, France.
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Lapidus, M.L. The Feynman-Kac formula with a Lebesgue-Stieltjes measure: An integral equation in the general case. Integr equ oper theory 12, 163–210 (1989). https://doi.org/10.1007/BF01195113
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DOI: https://doi.org/10.1007/BF01195113