Advertisement

Differential Equations

, Volume 54, Issue 4, pp 557–561 | Cite as

Itô Method for Proving the Feynman–Kac Formula for the Euclidean Analog of the Stochastic Schrödinger Equation

  • A. A. Loboda
Short Communications
  • 18 Downloads

Abstract

For a stochastic differential equation of the heat equation type, we obtain a Feynman–Kac formula to which the method of analytic continuation with respect to a parameter can be applied under certain assumptions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Diosi, L., Continuous quantum measurement and Itô formalism, Phys. Lett., 1988, vol. A129, no. 8/9, pp. 419–423.CrossRefGoogle Scholar
  2. 2.
    Smolyanov, O.G. and Truman, A., Schrödinger–Belavkin equations and associated Kolmogorov and Lindblad equations, Theoret. and Math. Phys., 1999. vol. 120, no. 2, pp. 973–984.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Belavkin, V.P. and Smolyanov, O.G., Feynman path integral corresponding to the stochastic Schrödinger equation, Dokl. Math., 1998, vol. 57, no. 3, pp. 430–434.zbMATHGoogle Scholar
  4. 4.
    Belavkin, V.P., A new wave equation for a continuous nondemolition measurement, Phys. Lett., 1989, vol. A140, no. 7/8, pp. 355–358.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Obrezkov, O.O. and Smolyanov, O.G., Relationship between the Itô–Schrödinger and Hudson–Parthasarathy equations, Dokl. Math., 2017, vol. 95, no. 1, pp. 87–91.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Obrezkov, O.O., Smolyanov, O.G., and Truman, A., The Generalized Chernoff theorem and randomized Feynman formula, Dokl. Math., 2005, vol. 71, no. 1, pp. 105–110.Google Scholar
  7. 7.
    Gough, J., Obrezkov, O.O., and Smolyanov, O.G., Randomized Hamiltonian Feynman integrals and Schrödinger–Itô stochastic equation, Izv. Math., 2005, vol. 69, no. 6, pp. 1081–1098.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Obrezkov, O.O., Stochastic Schrödinger equation with two-dimensional white noise, Russ. J. Math. Phys., 2002, vol. 9, no. 4, pp. 446–454.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Simon, B., Functional Integrals and Quantum Physics, New York: Academic, 1979.zbMATHGoogle Scholar
  10. 10.
    Smolyanov, O.G. and Shavgulidze, E.T., Kontinual’nye integraly (Path Integrals), Moscow: URSS, 2015.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations