# American Institute of Mathematical Sciences

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June  2014, 7(2): 361-379. doi: 10.3934/krm.2014.7.361

## A random cloud model for the Schrödinger equation

 1 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39 - 10117 Berlin

Received  October 2013 Revised  January 2014 Published  March 2014

The paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offspring. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.
Citation: Wolfgang Wagner. A random cloud model for the Schrödinger equation. Kinetic & Related Models, 2014, 7 (2) : 361-379. doi: 10.3934/krm.2014.7.361
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##### References:
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