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**

I. V. Sadovnichaya

Abstract

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The subject of this paper is the stochastic differential equation
of Schr\"odinger's type. In 1988 V. Belavkin (and L. Diosi in the most
important particular case) obtained the nonlinear Schr\"odinger 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=(-\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
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.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/98/982/98213t.htm

Last modified: June 17, 1998