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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)

Abstract

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].

Keywords

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

2010 Mathematics Subject Classification AMS

60H15 82C28 

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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

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