Recent Progress on the Dirichlet Forms Associated with Stochastic Quantization Problems

  • Rongchan ZhuEmail author
  • Xiangchan Zhu
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)


In this paper we present recent progress on the Dirichlet forms associated with stochastic quantization problems obtained in Röckner et al. (J Funct Anal, 272(10):4263–4303, 2017, [23]), Röckner et al. (Commun Math Phys, 352(3):1061–1090, 2017, [24]), Zhu and Zhu (Dirichlet form associated with the \(\varPhi _3^4\) model, 2017, [27]). In the two dimensional case we have obtained the equivalence of the two notions of solutions, the restricted Markov uniqueness and the uniqueness of martingale problem. In the three dimensional case we construct the Dirichlet form associated with the dynamical \(\varPhi ^4_3\) model obtained in Catellier and Chouk (Paracontrolled distributions and the 3-dimensional stochastic quantization equation, [6]), Hairer (Invent Math, 198:269–504, 2014, [14]), Mourrat and Weber (Global well-posedness of the dynamic \(\varPhi ^4_3\) model on the torus, [20].


\(\phi _3^4\) model Dirichlet form Regularity structures Paracontrolled distributions Space-time white noise Renormalisation 

2010 Mathematics Subject Classification AMS

60H15 82C28 


  1. 1.
    Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Relat. Field 89, 347–386 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albeverio, S., Röckner, M.: Dirichlet form methods for uniqueness of martingale problems and applications. Stochastic Analysis (Ithaca, NY, 1993). Proceedings of Symposia in Pure Mathematics, vol. 57, pp. 513–528. American Mathematical Socoiety, Providence (1995)CrossRefGoogle Scholar
  3. 3.
    Albeverio, S., Kondratiev, Y.G., Röckner, M.: Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs States. J. Funct. Anal. 149(2), 415–469 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Albeverio, S., Liang, S., Zegarlinski, B.: Remark on the integration by parts formula for the \(\Phi ^4_3\)-quantum field model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(1), 149–154 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  6. 6.
    Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation. arXiv:1310.6869
  7. 7.
    Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Feldman, J.: The \(\lambda \Phi ^4_ 3\) field theory in a finite volume. Commun. Math. Phys. 37, 93–120 (1974)Google Scholar
  9. 9.
    Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View. Springer, New York, Heidelberg, Berlin (1986)Google Scholar
  10. 10.
    Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn. Springer, New York (1987)CrossRefGoogle Scholar
  11. 11.
    Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. arXiv:1210.2684
  13. 13.
    Guerra, F., Rosen, J., Simon, B.: The \(P(\Phi )_2\) Euclidean quantum field theory as classical statistical mechanics. Ann. Math. 101, 111–259 (1975)Google Scholar
  14. 14.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198, 269–504 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs. arXiv:1511.06937v1
  16. 16.
    Jona-Lasinio, G., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101(3), 409–436 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin (1992)CrossRefGoogle Scholar
  18. 18.
    Mikulevicius, R., Rozovskii, B.: Martingale problems for stochasic PDE’s. Stochastic Partial Differential Equations: Six Perspectives. Mathematical Surveys and Monographs, vol. 64, pp. 243–325. American Mathematical Society, Providence (1999)CrossRefGoogle Scholar
  19. 19.
    Mourrat, J.-C., Weber, H.: Global well-posedness of the dynamic \(\Phi ^4\) model in the plane. arXiv:1501.06191v1
  20. 20.
    Mourrat, J.-C., Weber, H.: Global well-posedness of the dynamic \(\Phi ^4_3\) model on the torus. arXiv:1601.01234
  21. 21.
    Parisi, G., Wu, Y.S.: Perturbation theory without gauge fixing. Sci. Sinica 24(4), 483–496 (1981)MathSciNetGoogle Scholar
  22. 22.
    Röckner, M.: Specifications and Martin boundaries for \(P(\Phi )_2\)-random fields. Commun. Math. Phys. 106, 105–135 (1986)Google Scholar
  23. 23.
    Röckner, M., Zhu, R., Zhu, X.: Restricted Markov unqiueness for the stochastic quantization of \(P(\phi )_2\) and its applications (2015). arXiv:1511.08030; J. Funct. Anal. 272(10), 4263–4303 (2017)
  24. 24.
    Röckner, M., Zhu, R., Zhu, X.: Ergodicity for the stochastic quantization problems on the 2D-torus (2016). arXiv:1606.02102; Commun. Math. Phys. 352(3), 1061–1090 (2017)
  25. 25.
    Sickel, W.: Periodic spaces and relations to strong summability of multiple Fourier series. Math. Nachr. 124, 15–44 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Stein, E.M., Weiss, G.L.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  27. 27.
    Zhu, R., Zhu, X.: Dirichlet form associated with the \(\Phi _3^4\) model (2017). arXiv:1703.09987

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Institute of TechnologyBeijingChina
  2. 2.Department of MathematicsUniversity of BielefeldBielefeldGermany
  3. 3.School of ScienceBeijing Jiaotong UniversityBeijingChina

Personalised recommendations