Skip to main content
SpringerLink
Log in

Statistical analysis of some evolution equations driven by space-only noise

  • Published:
Statistical Inference for Stochastic Processes Aims and scope Submit manuscript

Abstract

We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations have been studied, little is known about inverse problems for these equations. We exploit the somewhat unusual structure of the observations coming from these equations that leads to an interesting interplay between classical and non-traditional statistical models. We derive several types of estimators for the drift and/or diffusion coefficients of these equations, and prove their relevant properties.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Casella G, Berger RL (2002) Statistical inference, 2nd edn. Cengage Learning & Wadsworth, Boston

    MATH  Google Scholar 

  • Cheng Z, Cialenco I, Gong R (2019) Bayesian estimations for diagonalizable bilinear SPDEs. Stoch Process Appl. https://doi.org/10.1016/j.spa.2019.03.020

    Article  MATH  Google Scholar 

  • Cialenco I (2018) Statistical inference for SPDEs: an overview. Stat Inference Stoch Process 21(2):309–329

    Article  MathSciNet  Google Scholar 

  • Cialenco I, Lototsky SV (2009) Parameter estimation in diagonalizable bilinear stochastic parabolic equations. Stat Inference Stoch Process 12(3):203–219

    Article  MathSciNet  Google Scholar 

  • Cialenco I, Huang Y (2017) A note on parameter estimation for discretely sampled SPDEs. Stoch Dyn. arXiv:1710.01649 (forthcoming)

  • Friedlander S, Glatt-Holtz N, Vicol V (2016) Inviscid limits for a stochastically forced shell model of turbulent flow. Ann Inst Henri Poincaré Probab Stat 52(3):1217–1247

    Article  MathSciNet  Google Scholar 

  • Frisch U (1995) Turbulence. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Glatt-Holtz N, Ziane M (2008) The stochastic primitive equations in two space dimensions with multiplicative noise. Discrete Contin Dyn Syst Ser B 10(4):801–822

    MathSciNet  MATH  Google Scholar 

  • Huebner M, Rozovskii BL (1995) On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE’s. Probab Theory Relat Fields 103(2):143–163

    Article  MathSciNet  Google Scholar 

  • Ibragimov IA, Has’minskiĭ RZ (1981) Statistical estimation: asymptotic theory. Applications of mathematics, vol 16. Springer, New York

    Book  Google Scholar 

  • Kakutani S (1948) On equivalence of infinite product measures. Ann Math 2(49):214–224

    Article  MathSciNet  Google Scholar 

  • Kim H-J, Lototsky SV (2017) Time-homogeneous parabolic Wick-Anderson model in one space dimension: regularity of solution. Stoch Partial Differ Equ Anal Comput 5(4):559–591

    MathSciNet  MATH  Google Scholar 

  • Klebaner FC (2005) Introduction to stochastic calculus with applications, 2nd edn. Imperial College Press, London

    Book  Google Scholar 

  • Lototsky SV, Rozovsky BL (2017) Stochastic partial differential equations. Universitext. Springer, Berlin

    Book  Google Scholar 

  • Lototsky SV, Rozovsky BL (2018) Stochastic evolution systems. Linear theory and applications to non-linear filtering, volume 89 of probability theory and stochastic modelling, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • Nourdin I, Peccati G, Swan Y (2014) Entropy and the fourth moment phenomenon. J Funct Anal 266(5):3170–3207

    Article  MathSciNet  Google Scholar 

  • Shiryaev AN (1996) Probability, volume 95 of graduate texts in mathematics, 2nd edn. Springer, New York

    Google Scholar 

  • Shubin MA (2001) Pseudodifferential operators and spectral theory, 2nd edn. Springer, Berlin

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the two anonymous referees, the associate editor and the editor for carefully reading the original manuscript, and their insightful comments and suggestions which improved greatly the final manuscript.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, 10 W 32nd Str, Building REC, Room 220, Chicago, IL, 60616, USA

    Igor Cialenco & Hyun-Jung Kim

  2. Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, KAP 104, Los Angeles, CA, 90089, USA

    Sergey V. Lototsky

Authors
  1. Igor Cialenco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyun-Jung Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergey V. Lototsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyun-Jung Kim.

Ethics declarations

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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

Cialenco, I., Kim, HJ. & Lototsky, S.V. Statistical analysis of some evolution equations driven by space-only noise. Stat Inference Stoch Process 23, 83–103 (2020). https://doi.org/10.1007/s11203-019-09205-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11203-019-09205-0

Keywords

Mathematics Subject Classification

Search

Navigation