Stochastic spikes and Poisson Approximation of one-dimensional stochastic differential equations with applications to continuously measured Quantum Systems

TL;DR

This paper investigates the limiting behavior of certain one-dimensional stochastic differential equations with a focus on quantum systems, demonstrating convergence to a Poisson process using classical probability methods.

Contribution

It extends previous work by providing a general convergence result to Poisson processes for these equations, relevant to quantum measurement analysis.

Findings

Proves convergence to a Poisson process under specified conditions

Extends prior results to a broader class of SDEs

Uses classical probabilistic tools for the analysis

Abstract

Motivated by the recent contribution \cite{BB17} we study the scaling limit behavior of a class of one-dimensional stochastic differential equations which has a unique attracting point subject to a small additional repulsive perturbation. Problems of this type appear in the analysis of continuously monitored quantum systems. We extend the results of \cite{BB17} and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools.