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Register a new userStrong uniqueness for Dirichlet operators related to stochastic quantization under exponential/trigonometric interactions on the two-dimensional torus
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We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.
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We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $exp (Phi)_{2}$-quantum field model or H{o}egh-Krohns model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.
The present paper is a continuation of our previous work on the stochastic quantization of the $exp(Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full $L^{1}$-regime $vertalphavert<sqrt{8pi}$ of the charge parameter $alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study the construction of the $Phi^3_3$-measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer (1988). This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the $Phi^3_3$-measure and show that it is singular with respect to the massive Gaussian free field. Moreover, we show that there exists a shifted measure with respect to which the $Phi^3_3$-measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors (2020), we establish non-normalizability of the $Phi^3_3$-measure. Due to the singularity of the $Phi^3_3$-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, we first construct a $sigma$-finite version of the $Phi^3_3$-measure and show that this measure is not normalizable. Furthermore, we prove that the truncated $Phi^3_3$-measures have no weak limit in a natural space, even up to a subsequence. We also study the dynamical problem. By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author (2018) and by the authors (2020), we prove almost sure global well-posedness of the hyperbolic $Phi^3_3$-model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, we introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.
We study a class of elliptic SPDEs with additive Gaussian noise on $mathbb{R}^2 times M$, with $M$ a $d$-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential $V$, convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in our previous paper on the topic, the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on the abstract Wiener space $L^2 (M)$. The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over $mathbb{T}^2$, and with exponential interaction over $mathbb{R}^2$ (known also as H{o}eg-Krohn or Liouville model in the literature). In particular for the exponential interaction case, the existence and uniqueness properties of solutions to the elliptic equation over $mathbb{R}^{2 + 2}$ is derived as well as the dimensional reduction for the values of the ``charge parameter $sigma = frac{alpha}{2sqrt{pi}} < sqrt{4 left( 8 - 4 sqrt{3} right) pi} simeq sqrt{4.23pi}$, for which the model has an Euclidean invariant probability measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space).
We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $Phi^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.
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