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Stochastic Quantization of the Φ3/3-Model
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Author: Oh, Tadahiro/ Okamoto, Mamoru/ Tolomeo, Leonardo Publisher: European Mathematical Society ISBN: 9783985470853 Cover: PAPERBACK Date: 2025年05月 DESCRIPTION The authors study the construction of the Φ 3/3--measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer [J. Statist. Phys. 50 (1988), no. 3-4, 657-687]. This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, the authors prove normalizability of the Φ 3/3-measure and show that it is singular with respect to the massive Gaussian free field. Moreover, the authors show that there exists a shifted measure with respect to which the Φ 3/3-measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery tthey have introduced, , the authors establish non-normalizability of the Φ 3/3-measure. Due to the singularity of the Φ 3/3-measure with respect to the massive Gaussian free field, this non-normalizability part poses a particular challenge as compared to our previous works. In order to overcome this issue, the authors first construct a σ-finite version of the Φ 3/3-measure and show that this measure is not normalizable. Furthermore, the authors prove that the truncated Φ 3/3-measures have no weak limit in a natural space, even up to a subsequence. The authors also study the dynamical problem for the canonical stochastic quantization of the Φ 3/3measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic Φ 3/3-model). By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author [J. Eur. Math. Soc. 26 (2024), no. 3, 817-874] and by the authors [Mem. Amer. Math. Soc. 304 (2024), no. 1529], the authors prove almost sure global well-posedness of the hyperbolic Φ 3/3-model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, they introduce a new, conceptually simple and straightforward approach, where they directly work with the (truncated) Gibbs measure, using the Boue-Dupuis variational formula and ideas from the theory of optimal transport. TABLE OF CONTENTS 1 Introduction 2 Notations and basic lemmas 3 Construction of the Φ 3/3-measure in the weakly nonlinear regime 4 Non-normalizability in the strongly nonlinear regime 5 Local well-posedness 6 Invariant Gibbs dynamics 最近チェックした商品
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