Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics

Bringmann, Bjoern

doi:10.48550/arxiv.2009.04616

Cited by 10 publications

(87 citation statements)

References 38 publications

(95 reference statements)

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confidence: 99%

“…for a power-type nonlinearity [33,34,35,21,22,56,50,49,70,51,13,62] and for trigonometric and exponential nonlinearities [54,57,55]. We also mention the works [61,53,52,13] on nonlinear wave equations with rough random initial data. In [34], by combining the paracontrolled calculus, originally introduced in the parabolic setting [30,16,45], with the multilinear harmonic analytic approach, more traditional in studying dispersive equations, Gubinelli, Koch, and the first author studied the quadratic SNLW (1.10) (without the 7 Some of the works mentioned below are on SNLW without damping.…”

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“…In the same work, we also constructed the invariant Gibbs dynamics for the associated (canonical) stochastic quantization equation. See also [51,12,13] for the defocusing case (σ < 0). Note that when β = 0, the defocusing Hartree Φ 4 3 -measure reduces to the usual Φ 4 3 -measure.…”

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“…damping). The paracontrolled approach in the wave setting was also used in our previous work [51] and was further developed by Bringmann [13]. In order to prove local well-posedness of the hyperbolic Φ 3 3 -model (1.10), we also follow the paracontrolled approach, in particular combining the analysis in [34,51].…”

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Stochastic quantization of the $Φ^3_3$-model

Oh¹,

Okamoto²,

Tolomeo³

2021

Preprint

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We 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 (1988). This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the Φ 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 Φ 3 3 -measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors (2020) and the first and third authors with K. Seong (2020), we 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, we first construct a σ-finite version of the Φ 3 3 -measure and show that this measure is not normalizable. Furthermore, we prove that the truncated Φ 3 3 -measures have no weak limit in a natural space, even up to a subsequence. We also study the dynamical problem for the canonical stochastic quantization of the Φ 3 3 -measure, 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 3model). 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 Φ 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 Boué-Dupuis variational formula and ideas from theory of optimal transport.

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Stochastic quantization of the $Φ^3_3$-model

Oh¹,

Okamoto²,

Tolomeo³

2021

Preprint

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“…In recent years, there has been tremendous interest in random dispersive equations. In this introduction, we only discuss selected works in this field and refer the reader to the surveys [BOP19,NS19], the introduction of [Bri20], and the original works [BOP15, Bou94, Bou96, Bri18, Bri20, BT08a, BT08b, CG19, DH19, DH21, DLM20, DNY19, DNY20, DNY21, GKO18, KLS20, KM19, KMV20, LM14, NORBS12, OOT20, OOT21, OST21, ST21, Tzv15].…”

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The wave maps equation and Brownian paths

Bringmann¹,

Lührmann²,

Staffilani³

2021

Preprint

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Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

Tolomeo

2020

Commun. Math. Phys.

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In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for $$d=1$$ d = 1 and the beam equation for $$d\le 3$$ d ≤ 3 . We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.

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Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics

Cited by 10 publications

References 38 publications

Stochastic quantization of the $Φ^3_3$-model

Stochastic quantization of the $Φ^3_3$-model

The wave maps equation and Brownian paths

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

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