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Fundam. Prikl. Mat., 1998, Volume 4, Issue 2, Pages 659–667 (Mi fpm323)  

This article is cited in 2 scientific papers (total in 2 papers)

About one representation of the solution of Schrödinger stochastic equation by means of an integral over the Wiener measure

I. V. Sadovnichaya

M. V. Lomonosov Moscow State University

Abstract: The subject of this paper is the stochastic differential equation of Schrödinger's type. In 1988 V. Belavkin (and L. Diosi in the most important particular case) obtained the nonlinear Schrödinger equation, which describes the evolution of the quantum system under the continuous measurement. In the first part of this paper we analyze the following stochastic equation:
$$ \mathop{id}\psi=(-\Delta/2-i\lambda/4\cdot\|q\|^2+v(q))\psi dt +i\sqrt{\lambda/2}q\psi dB, $$
which is the particular case of Belavkin equation, and present an explicit formula of diffusion process — the solution of this equation. (This result was announced in the paper [1].) This solution is the integral over Wiener measure. In the second part it is represented as the limit of the suitable sequnce of finite-dimensional integrals, which are used in the definition of Feynman integral.

Full text: PDF file (266 kB)

Bibliographic databases:
UDC: 517.948+513.8.519.4
Received: 01.05.1997

Citation: I. V. Sadovnichaya, “About one representation of the solution of Schrödinger stochastic equation by means of an integral over the Wiener measure”, Fundam. Prikl. Mat., 4:2 (1998), 659–667

Citation in format AMSBIB
\Bibitem{Sad98}
\by I.~V.~Sadovnichaya
\paper About one representation of the~solution of Schr\"odinger stochastic equation by means of an~integral over the~Wiener measure
\jour Fundam. Prikl. Mat.
\yr 1998
\vol 4
\issue 2
\pages 659--667
\mathnet{http://mi.mathnet.ru/fpm323}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1801180}
\zmath{https://zbmath.org/?q=an:0966.60055}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Liu HePing, Wang YingZhan, “Wiener Measure for Heisenberg Group”, Sci. China-Math., 57:8 (2014), 1605–1614  crossref  mathscinet  zmath  isi
    2. A. A. Loboda, “The Doss Method for the Stochastic Schrödinger–Belavkin Equation”, Math. Notes, 106:2 (2019), 303–307  mathnet  crossref  crossref  isi  elib
  • Фундаментальная и прикладная математика
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