Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation

Oh, Tadahiro; Okamoto, Mamoru; Tzvetkov, Nikolay

doi:10.48550/arxiv.2206.00728

Cited by 1 publication

(1 citation statement)

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“…Here, N (u) denotes a nonlinearity which may be of a power-type [34,35,36,65,57,80,58,13,59,70,15] and trigonometric and exponential nonlinearities [63,66,64]. We also mention the works [69,61,60,76,71] on the (deterministic) nonlinear wave equations (1.4) with rough random initial data and [24,25,56] on SNLW with more singular (both in space and time) noises. We point out that the only known well-posedness result up to date for singular SNLW posed on an unbounded domain is the work [80] by the second author, where he established pathwise global well-posedness of the cubic SNLW on R 2 with an additive space-time white noise forcing:…”

Section: As For the Construction Of The Limiting φ K+1mentioning

confidence: 99%

Hyperbolic $P(Φ)_2$-model on the plane

Oh¹,

Tolomeo²,

Wang³

et al. 2022

Preprint

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We study the hyperbolic Φ k+1 2 -model on the plane. By establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a Φ k+1 2 -measure on the plane as a limit of the Φ k+1 2 -measures on large tori. We then study the canonical stochastic quantization of the Φ k+1 2 -measure on the plane thus constructed, namely, we study the defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise (= the hyperbolic Φ k+1 2 -model) on the plane. In particular, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the hyperbolic Φ k+1 2 -model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic Φ 3 3 -model on the three-dimensional torus by the first two authors and Okamoto ( 2021), and to study convergence of the socalled enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic Φ k+1 2 -model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting Φ k+1 2 -measure on the plane under the dynamics of the parabolic Φ k+1 2 -model.

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