Existence, uniqueness and approximation for stochastic Schrodinger equation: the Poisson case

TL;DR

This paper rigorously establishes the existence and uniqueness of solutions for stochastic Schrödinger equations in the Poisson case, using random measure theory and connecting the model to physical discrete-time systems.

Contribution

It provides the first rigorous mathematical proof of existence and uniqueness for the Poisson case of stochastic Schrödinger equations, linking the model to physical systems.

Findings

Proved existence and uniqueness of solutions.

Connected stochastic models to discrete physical systems.

Used random measure theory for rigorous justification.

Abstract

In quantum physics, recent investigations deal with the so-called "quantum trajectory" theory. Heuristic rules are usually used to give rise to "stochastic Schrodinger equations" which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical justification: notion of solution, existence, uniqueness, justification... In this article, we concentrate on a particular case: the Poisson case. Random measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of a solution for the associated stochastic equation. Furthermore, the stochastic model is physically justified by proving that the solution can be obtained as a limit of a concrete discrete time physical model.