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Global Solutions to Elliptic and Parabolic \({\Phi^4}\) Models in Euclidean Space

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Abstract

We prove the existence of global solutions to singular SPDEs on \({\mathbb{R}^{\rm d}}\) with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions d = 4, 5 and in the parabolic setting for d = 2, 3. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the \({\Phi^{4}_d}\) Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the \({\Phi^{4}_{d-2}}\) Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism.

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References

  1. Albeverio, S., Kusuoka S.: The invariant measure and the flow associated to the \({\Phi^{4}_{3}}\) -quantum field model. arXiv:1711.07108, November (2017)

  2. Aliprantis C.D., Border K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide. 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  3. Bahouri H., Chemin J.-Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  4. Badiale M., Serra E.: Semilinear Elliptic Equations for Beginners. Universitext. Springer, London (2011) (Existence results via the variational approach)

    Book  MATH  Google Scholar 

  5. Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation. Ann. Probab. 46(5), 2621–2679 (2018)

  6. Debussche A., de Moor S., Hofmanová M.: A regularity result for quasilinear stochastic partial differential equations of parabolic type. SIAM J. Math. Anal. 47(2), 1590–1614 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Evans L.C.: Partial Differential Equations. 2nd edn. American Mathematical Society, New York City (2010)

    MATH  Google Scholar 

  8. Furlan, M., Gubinelli, M.: Weak universality for a class of 3d stochastic reaction-diffusion models. arXiv:1708.03118 [math] (2017)

  9. Gubinelli M., Imkeller P., Perkowski N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6,75 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gubinelli M., Perkowski N.: KPZ reloaded. Commun. Math. Phys. 349(1), 165–269 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Hairer M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Hairer M., Labbé C.: A simple construction of the continuum parabolic Anderson model on R 2. Electron. Commun. Probab. 20(43), 11 (2015)

    MATH  Google Scholar 

  13. Hairer M., Labbé C.: Multiplicative stochastic heat equations on the whole space. J. Eur. Math. Soc. (JEMS) 20(4), 1005–1054 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klein A., Fernando Perez J.: Supersymmetry and dimensional reduction: a non-perturbative proof. Phys. Lett. B 125(6), 473–475 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  15. Klein A., Landau L.J., Fernando Perez J.: Supersymmetry and the Parisi–Sourlas dimensional reduction: a rigorous proof. Commun. Math. Phys. 94(4), 459–482 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Kupiainen A.: Renormalization group and stochastic PDEs. Annales Henri Poincaré 17(3), 497–535 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I. (1968)

  18. Mourrat J.-C., Weber H.: The dynamic \({\Phi^{4}_{3}}\) model comes down from infinity. Commun. Math. Phys 356(3), 673–753 (2017)

    Article  ADS  MATH  Google Scholar 

  19. Mourrat J.-C., Weber H.: Global well-posedness of the dynamic \({\Phi^4}\) model in the plane. Ann. Probab. 45(4), 2398–2476 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mourrat, J.-C., Weber, H., Xu, W.: Construction of \({\Phi^{4}_{3}}\) diagrams for pedestrians. arXiv:1610.08897 [math-ph] (2016)

  21. Otto, F., Weber, H.: Quasilinear SPDEs via rough paths. arXiv:1605.09744 [math] (2016)

  22. Parisi G., Sourlas N.: Random magnetic fields, supersymmetry, and negative dimensions. Phys. Rev. Lett. 43(11), 744–745 (1979)

    Article  ADS  Google Scholar 

  23. Parisi G., Wu Y.S.: Perturbation theory without gauge fixing. Scientia Sinica. Zhongguo Kexue 24(4), 483–496 (1981)

    MathSciNet  Google Scholar 

  24. Rodino L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific Publishing Co., Inc., River Edge (1993)

    Book  MATH  Google Scholar 

  25. Simon J.: Compact sets in the space \({L^p(0, T; B)}\) . Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schmeisser H.-J., Triebel H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987)

    MATH  Google Scholar 

  27. Triebel H.: Theory of Function Spaces II. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  28. Triebel H.: Theory of Function Spaces III. Springer, Berlin (2006)

    MATH  Google Scholar 

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Acknowledgements

MG is partially supported by the German Research Foundation (DFG) via CRC 1060.

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Authors and Affiliations

  1. Hausdorff Center for Mathematics and Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Massimiliano Gubinelli

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501, Bielefeld, Germany

    Martina Hofmanová

Authors
  1. Massimiliano Gubinelli
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  2. Martina Hofmanová
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Correspondence to Martina Hofmanová.

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Communicated by M. Hairer

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Gubinelli, M., Hofmanová, M. Global Solutions to Elliptic and Parabolic \({\Phi^4}\) Models in Euclidean Space. Commun. Math. Phys. 368, 1201–1266 (2019). https://doi.org/10.1007/s00220-019-03398-4

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