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Renormalization Group Approach to SDEs with Nonlinear Diffusion Terms

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Abstract

Renormalization Group (RG) method has been recognized as an unified and effective approach for singularly perturbed problems in many fields, such as oscillation and boundary layer problems in ordinary differential equations (ODEs), center manifolds in dynamical systems, and long-time asymptotic behavior in partial differential equations (PDEs), etc. In this paper, we are going to investigate the application of RG method to the asymptotic behavior of a class of stochastic differential equation with multiplicative noise. We will first show how to formulate the RG equation and the approximate solution. Then, the rigorous estimation of the error between the approximate solution and the real one will be presented.

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Notes

  1. \(a \wedge b :=\min \{a,b\}\).

References

  1. Blömker, D., Gugg, C., Maier-Paape, S.: Stochastic Navier–Stokes equation and renormalization group theory. Phys. D 173, 137–152 (2002)

    Article  MathSciNet  Google Scholar 

  2. Borli, I., Sutera, A.: Stochastic perturbation in meteorology. Waves Random Media 10, R1–R30 (2000)

    Article  MathSciNet  Google Scholar 

  3. Boyanovsky, D., De-Vega, H.J.: Dynamical renormalization group approach to relaxation in quantum field theory. Ann. Phys. 307, 335–371 (2003)

    Article  MathSciNet  Google Scholar 

  4. Bulla, R., Costi, T., Pruschke, T.: The numerical renormalization group method for quantum impurity systems. Rev. Modern Phys. 80, 395–450 (2008)

    Article  Google Scholar 

  5. Chen, L.Y., Goldenfeld, N., Oono, Y.: Renormalization group theory for global asymptotic analysis. Phys. Rev. Lett. 73, 1311–1315 (1994)

    Article  MathSciNet  Google Scholar 

  6. Chen, L.Y., Goldenfeld, N., Oono, Y.: Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. E 54, 376–394 (1996)

    Article  Google Scholar 

  7. Chiba, H.: Approximation of center manifolds on the renormalization group method. J. Maths. Phys. 49, 102703 (2008)

    Article  MathSciNet  Google Scholar 

  8. Chiba, H.: Extension and unification of singular perturbation methods for ODEs based on the renormalization group method. SIAM J. Appl. Dyn. Syst. 8, 1066–1115 (2009)

    Article  MathSciNet  Google Scholar 

  9. Da-Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  10. Dunbar, S.R.: Mathematical Modeling in Economics and Finance: Probability, Stochastic Processes, and Differential Equations. An Imprint of the American Mathematical Society, volume: 49, (2019)

  11. Franzke, C.L.E., O’Kane, T.J.: Nonlinear and Stochastic Climate Dynamics. Cambridge University Press, Cambridge (2017)

  12. Frasca, M.: Renormalization group methods in quantum optics. Phys. Rev. A 56, 1548–1552 (1997)

    Article  Google Scholar 

  13. Galusinski, C.: A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence, M2AN Math. Model. Numer. Anal. 34, 109–125 (2000)

    Article  MathSciNet  Google Scholar 

  14. Glatt-Holtz, N., Ziane, M.: Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete Contin. Dyn. Syst. 26, 1241–1268 (2010)

    Article  MathSciNet  Google Scholar 

  15. Glatt-Holtz, N., Ziane, M.: Strong pathwise of solutions to stochastic Navier–Stokes system. Adv. Differential Equations 14, 567–600 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Hausenblas, E., Seidler, J.: A note on maximal inequality for stochastic convolutions. Czech. Math. J. 51, 785–790 (2001)

    Article  MathSciNet  Google Scholar 

  17. Holmes, M.H.: Introduction to Perturbation Methods, 2nd edn. Springer, New York (2013)

    Book  Google Scholar 

  18. Kabanov, YuM, Pergamenshchikov, S.M.: Singular perturbations of stochastic differential equations. Mat. Sb. 181, 1170–1182 (1990)

    MATH  Google Scholar 

  19. Li, W.L., Shi, S.Y.: Singular perturbed renormalization group theory and its application to oscillatory problems. Discrete Contin. Dyn. Syst. B 23, 1819–1833 (2018)

    Article  MathSciNet  Google Scholar 

  20. Moise, I., Temam, R.: Renormalization group method. Application to Navier–Stokes equation. Discrete Contin. Dyn. Syst. 6, 191–210 (2000)

    Article  MathSciNet  Google Scholar 

  21. Moise, I., Ziane, M.: Renormalization group method. Applications to partial differential equations. J. Dynam. Differ. Eqs. 13, 275–321 (2001)

    Article  MathSciNet  Google Scholar 

  22. Nayfeh, A.L.: Pertuabation Methods. Wiley, New York (1973)

    Google Scholar 

  23. Veysey II, J., Goldenfeld, N.: Simple viscous flows: from boundary layers to the renormalization group. Rev. Modern Phys. 79, 883–927 (2007)

    Article  MathSciNet  Google Scholar 

  24. Wilkinson, D.J.: Stochastic Modelling for Systems Biology. Chapman & Hall/CRC, Boca Raton (2006)

    Book  Google Scholar 

  25. Ziane, M.: On a certain renormalization group method. J. Math. Phys. 41, 3290–3299 (2003)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We sincerely thank referee for helpful suggestions and comments.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, P. R. China

    Shiduo Qu & Wenlei Li

  2. School of Mathematics & State key laboratory of automotive simulation and control, Jilin University, Changchun, 130012, P. R. China

    Shaoyun Shi

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  1. Shiduo Qu
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  2. Wenlei Li
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  3. Shaoyun Shi
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Correspondence to Shaoyun Shi.

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This work is supported by NSFC grant (No. 11771177, 11301210), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20), NSF grant (No. 20190201132JC) and ESF grant (No. JJKH20170776KJ) of Jilin, China.

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Qu, S., Li, W. & Shi, S. Renormalization Group Approach to SDEs with Nonlinear Diffusion Terms. Mediterr. J. Math. 18, 183 (2021). https://doi.org/10.1007/s00009-021-01821-6

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  • DOI: https://doi.org/10.1007/s00009-021-01821-6

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