Strong Uniqueness of Dirichlet Operators Related to Stochastic Quantization Under Exponential Interactions in One-Dimensional Infinite Volume

  • Hiroshi KawabiEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)


In this survey paper, we discuss strong uniqueness of Dirichlet operators related to stochastic quantization under exponential (and polynomial) interactions in one-dimensional infinite volume based on joint works with Sergio Albeverio and Michael Röckner (Albeverio et al., J Funct Anal 262:602–638, 2012, [4], Kawabi and Röckner, J Funct Anal 242:486–518, 2007, [11]). We also raise an open problem.


Strong uniqueness \(L^{p}\)-uniqueness Essential self-adjointness Dirichlet operator Stochastic quantization Gibbs measure Path space SPDE 

2010 AMS Classification Numbers

35R15 35R60 46N50 47D07 



The author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 26400134.


  1. 1.
    Albeverio, S., Høegh-Krohn, R.: Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non-polynomial interactions. Commun. Math. Phys. 30, 171–200 (1973)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albeverio, S., Høegh-Krohn, R.: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time. J. Funct. Anal. 16, 39–82 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Relat. Fields 89, 347–386 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Albeverio, S., Kawabi, H., Röckner, M.: Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions. J. Funct. Anal. 262, 602–638 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Albeverio, S., Ma, Z.M., Röckner, M.: Quasi regular Dirichlet forms and the stochastic quantization problem. Festschrift Masatoshi Fukushima, 27–58, Interdisciplinary Mathematical Sciences, vol. 17. World Scientific Publishing, Hackensack (2015)Google Scholar
  6. 6.
    Eberle, A.: Uniqueness and non-uniqueness of singular diffusion operators. Lecture Notes in Mathematics, vol. 1718. Springer, Berlin (1999)Google Scholar
  7. 7.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6, 75 p. (2015)Google Scholar
  8. 8.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198, 269–504 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Iwata, K.: Reversible measures of a \(P(\phi )_{1}\)- time evolution. In: Itô, K., Ikeda, N. (eds.), Probabilistic Methods in Mathematical Physics: Proceedings of Taniguchi Symposium, pp. 195–209. Kinokuniya (1985)Google Scholar
  10. 10.
    Iwata, K.: An infinite dimensional stochastic differential equation with state space \(C({ R})\). Probab. Theory Relat. Fields 74, 141–518 (1987)Google Scholar
  11. 11.
    Kawabi, H., Röckner, M.: Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via an SPDE approach. J. Funct. Anal. 242, 486–518 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lőrinczi, J., Hiroshima, F., Betz, V.: Feynman-Kac-type theorems and Gibbs measures on path space. De Gruyter Studies in Mathematics, vol. 34. Walter de Gruyter Co., Berlin (2011)Google Scholar
  13. 13.
    Mourrat, J.-C., Weber, H.: Global well-posedness of the dynamic \(\Phi ^{4}\) model in the plane. Ann. Probab. 45, 2398–2476 (2017)Google Scholar
  14. 14.
    Ondreját, M.: Uniqueness for stochastic evolution equations in Banach spaces. Dissertaiones Math. (Rozprawy Mat.) 426 (2004), 63 pGoogle Scholar
  15. 15.
    Röckner, M., Zhu, R., Zhu, X.: Restricted Markov uniqueness for the stochastic quantization of \(P(\Phi )_{2}\) and its applications. J. Funct. Anal. 272, 4263–4303 (2017)Google Scholar
  16. 16.
    Rosen, J., Simon, B.: Fluctuations in \(P(\phi )_{1}\) processes. Ann. Probab. 4, 155–174 (1976)Google Scholar
  17. 17.
    Shiga, T.: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can. J. Math. 46, 415–437 (1994)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Science, Department of MathematicsOkayama UniversityKita-ku, OkayamaJapan

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