Skip to main content
SpringerLink
Log in

A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein–Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  2. Da Prato, G., Kwapieň, S., Zabczyk, J.: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23(1), 1–23 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Freidlin, M.: Some remarks on the Smoluchowski–Kramers approximation. J. Stat. Phys. 117, 617–634 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer Series in Synergetics, 3rd edn. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  5. Guillouzic, S., L’Heureux, I., Longtin, A.: Small delay approximation of stochastic delay differential equations. Phys. Rev. E 59(4), 3970–3982 (1999)

    Article  ADS  MATH  Google Scholar 

  6. Heil, T., Fischer, I., Elsäßer, W., Krauskopf, B., Green, K., Gavrielides, A.: Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms. Phys. Rev. E 67, 066214 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  7. Hottovy, S., Volpe, G., Wehr, J.: Thermophoresis of Brownian particles driven by coloured noise. EPL (Europhys. Lett.) 99, 60002 (2012)

    Article  ADS  Google Scholar 

  8. Hottovy, S., McDaniel, A., Volpe, G., Wehr, J.: The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. Commun. Math. Phys. 336(3), 1259–1283 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Janson, S.: Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997). https://doi.org/10.1017/CBO9780511526169

    Book  Google Scholar 

  10. Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960)

    Article  Google Scholar 

  11. Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  12. Kupferman, R., Pavliotis, G.A., Stuart, A.M.: Itô versus Stratonovich white-noise limits for systems with inertia and colored multiplicative noise. Phys. Rev. E 70, 036120 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  13. Kurtz, T.G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19(3), 1035–1070 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lenstra, D., Verbeek, B.H., den Boef, A.J.: Coherence collapse in single-mode semiconductor lasers due to optical feedback. IEEE J. Quantum Electron. QE–21(6), 674–679 (1985)

    Article  ADS  Google Scholar 

  15. Lim, S.H., Wehr, J.: Homogenization for a class of generalized Langevin equations with an application to thermophoresis. J. Stat. Phys. (2017). https://doi.org/10.1007/s10955-018-2192-9

  16. Longtin, A.: Stochastic delay-differential equations. In: Atay, F.M. (ed.) Complex Time-Delay Systems, pp. 177–195. Springer, Berlin (2010)

    Google Scholar 

  17. Mao, X.: Stochastic Differential Equations and Applications. Woodhead Publishing, Cambridge (2007)

    MATH  Google Scholar 

  18. McDaniel, A., Mahalov, A.: Stochastic differential equation model for spontaneous emission and carrier noise in semiconductor lasers. IEEE J. Quantum Electron. 54(1), 2000206 (2018)

    Article  Google Scholar 

  19. McDaniel, A., Duman, O., Volpe, G., Wehr, J.: An SDE approximation for stochastic differential delay equations with state-dependent colored noise. Markov Process. Relat. 22(3), 595–628 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Mijalkov, M., McDaniel, A., Wehr, J., Volpe, G.: Engineering sensorial delay to control phototaxis and emergent collective behaviors. Phys. Rev. X 6(1), 011008 (2016)

    Google Scholar 

  21. Mulet, J., Mirasso, C.R.: Numerical statistics of power dropouts based on the Lang–Kobayashi model. Phys. Rev. E 59(5), 5400–5405 (1999)

    Article  ADS  Google Scholar 

  22. Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton, N.J. (1967)

    MATH  Google Scholar 

  23. Øksendal, B.: Stochastic Differential Equations, 6th edn. Springer, Berlin (2003). An Introduction with Applications

    Book  MATH  Google Scholar 

  24. Pavliotis, G.A., Stuart, A.M.: Analysis of white noise limits for stochastic systems with two fast relaxation times. Multiscale Model. Simul. 4(1), 1–35 (2005). https://doi.org/10.1137/040610507

    Article  MathSciNet  MATH  Google Scholar 

  25. Pavliotis, G.A., Stuart, A.M.: Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics, vol. 53. Springer, New York (2008)

    MATH  Google Scholar 

  26. Peligrad, M., Utev, S.: A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33(2), 798–815 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pesce, G., McDaniel, A., Hottovy, S., Wehr, J., Volpe, G.: Stratonovich-to-Itô transition in noisy systems with multiplicative feedback. Nat. Commun. 4, 2733 (2013). https://doi.org/10.1038/ncomms3733

    Article  ADS  Google Scholar 

  28. Protter, P.E.: Stochastic Integration and Differential Equations, Stochastic Modelling and Applied Probability, 2nd edn. Springer, Berlin (2005)

    Book  Google Scholar 

  29. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (1999)

    Google Scholar 

  30. Tian, T., Burrage, K., Burrage, P.M., Carletti, M.: Stochastic delay differential equations for genetic regulatory networks. J. Comput. Appl. Math. 205, 696–707 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Torcini, A., Barland, S., Giacomelli, G., Marin, F.: Low-frequency fluctuations in vertical cavity lasers: experiments versus Lang–Kobayashi dynamics. Phys. Rev. A 74(6), 063801 (2006)

    Article  ADS  Google Scholar 

  32. Volpe, G., Wehr, J.: Effective drifts in dynamical systems with multiplicative noise: a review of recent progress. Rep. Prog. Phys. 79(5), 053901 (2016)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

A.M. and J.W. were partially supported by the NSF Grants DMS 1009508 and DMS 0623941.

Author information

Authors and Affiliations

  1. Department of Mathematics, United States Naval Academy, Annapolis, MD, 21401, USA

    Scott Hottovy

  2. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Austin McDaniel & Jan Wehr

  3. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Austin McDaniel

Authors
  1. Scott Hottovy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Austin McDaniel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Wehr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Austin McDaniel.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hottovy, S., McDaniel, A. & Wehr, J. A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise. J Stat Phys 175, 19–46 (2019). https://doi.org/10.1007/s10955-019-02242-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-019-02242-2

Keywords

Search

Navigation