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].
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Zhu, R., Zhu, X. (2018). Recent Progress on the Dirichlet Forms Associated with Stochastic Quantization Problems. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_39
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