FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1998, VOLUME 4, NUMBER 2, PAGES 659-667

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

Abstract

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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:

$id \psi =\left(-\Delta /2-i\lambda /4\cdot \|q\|^2+v\left(q\right)\right)\psi dt+ i\sqrt \left\{\lambda /2\right\}q\psi dB,$

which is the particular case of Belavkin equation, and present an explicit formula of diffusion process -- the solution of this equation. This 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.

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