Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

Oh, Tadahiro; Wang, Yuzhao; Zine, Younes

doi:10.1007/s40072-022-00237-x

Cited by 7 publications

(3 citation statements)

References 51 publications

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“…Indeed, local well posedness for equation (SDNLW) has been proven in [20] on T 2 , in [40,35] on R 2 , and in [34] on a general 2-dimensional compact manifold. A series of 3-dimensional results have been proven in [21] and [33] for the equation with quadratic nonlinearity, in [32,3,36] for cubic nonlinearities under the addition of some smoothing in the equation, and finally in [4] for the canonical stochastic quantisation equation for the Φ 4 3 measure. When an invariant measure is available, often global well-posedness for a.e.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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

Tolomeo

2020

Commun. Math. Phys.

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“…where the noise ζ is primarily taken to be a space-time white noise ξ. 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.…”

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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“…Indeed, although, in the polynomial case, the well-posedness theory for (1.2) has been known for some time since the work of Da Prato and Debussche [14], it is only until recently that well-posedness for (1.1) is well understood. See [17,18,19,30,31,32,37]. In the case of a sine nonlinearity; namely, for the so-called sine-Gordon model, the well-posedness theories for the wave and heat equations are also very recent [21,34,35,12].…”

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confidence: 99%

Smoluchowski-Kramers approximation for the singular stochastic wave equations in two dimensions

Zine¹

2022

Preprint

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We study a family of nonlinear damped wave equations indexed by a parameter ε > 0 and forced by a space-time white noise on the two dimensional torus, with polynomial and sine nonlinearities. We show that as ε → 0, the solutions to these equations converge to the solution of the corresponding two dimensional stochastic quantization equation. In the sine nonlinearity case, the convergence is proven over arbitrary large times, while in the polynomial case, we prove that this approximation result holds over arbitrary large times when the parameter ε goes to zero even with a lack of suitable global well-posedness theory for the corresponding wave equations.

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Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

Cited by 7 publications

References 51 publications

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

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

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

Smoluchowski-Kramers approximation for the singular stochastic wave equations in two dimensions

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