About one representation of the~solution of Schr\"odinger stochastic equation by means of an~integral over the~Wiener measure

I. V. Sadovnichaya

Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 659-667.

Résumé

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.