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Stochastic Spikes and Poisson Approximation of One-Dimensional Stochastic Differential Equations with Applications to Continuously Measured Quantum Systems

  • Martin Kolb
  • Matthias LiesenfeldEmail author
Article

Abstract

Motivated by the recent contribution (Bauer and Bernard in Annales Henri Poincaré 19:653–693, 2018), 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 Bauer and Bernard (Annales Henri Poincaré 19:653–693, 2018) and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools.

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References

  1. 1.
    Bauer, M., Bernard, D., Tilloy, A.: Zooming in on quantum trajectories. J. Phys. A Math. Theor. 49(10), 10LT01 (2016).  https://doi.org/10.1088/1751-8113/49/10/10LT01
  2. 2.
    Bauer, M., Bernard, D.: Stochastic spikes and strong noise limits of stochastic differential equations. Annales Henri Poincaré 19, 653–693 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bass, R.F.: Stochastic Processes, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2011)Google Scholar
  4. 4.
    Bernardin, C., Chetrite, R., Chhaibi, R., Najnudel, J., Pellegrini, C.: Spiking and collapsing in large noise limits of SDE’s, manuscript, arxiv:1810.05629 (2018)
  5. 5.
    Borkovec, M., Klüppelberg, C.: Extremal behavior of diffusion models in finance. Extremes 1, 47–80 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brown, M.: Error bound for exponential approximation of geometric convolutions. Ann. Probab. 18, 1388–1402 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Durrett, R.: Probability: Theory and Examples, 3rd edn. Thomson, Brooks Cole, Belmont (2005)Google Scholar
  8. 8.
    Feller, W.: The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468–519 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fitzsimmons, P.J., Pitman, J.: Kac’s moment formula and the Feyman–Kac formula for additive functionals of a Markov process. Stoch. Process. Their Appl. 79, 117–134 (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lai, T.L.: Uniform Tauberian theorems and their applications to renewal theory and first passage problems. Ann. Probab. 4, 628–642 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Löcherbach, E., Loukianova, D., Loukianov, O.: Polynomial bound in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times. Annales de l’Institut Poincaré et Statistique 47, 425–449 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mandl, P.: Analytical Treatment of One-Dimensional Markov Processes. Springer, Berlin (1968)zbMATHGoogle Scholar
  13. 13.
    Pinsky, R.: Positive Harmonic Functions and Diffusion. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  14. 14.
    Tilloy, A., Bauer, M., Bernard, D.: Spikes in quantum trajectories. Phys. Rev. A 92(5), 052111 (2015).  https://doi.org/10.1103/PhysRevA.92.052111
  15. 15.
    Weidmann, J.: Lineare Operatoren in Hilberträumen: Teil II: Anwendungen. Vieweg+Teubner Verlag, Berlin (2013)zbMATHGoogle Scholar
  16. 16.
    Yannaros, N.: Poisson approximation for random sums of Bernoulli random variables. Stat. Probab. Lett. 11, 161–165 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany

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