On the unique ergodicity for a class of 2 dimensional stochastic wave equations

Forlano, Justin; Tolomeo, Leonardo

doi:10.48550/arxiv.2102.09075

Cited by 3 publications

(3 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This was further developed in [71] to prove ergodicity of the hyperbolic Φ 4 2 -model, namely (1.13) on T 2 with N (u) = u 3 . See also [26] by the third author and Forlano on the asymptotic Feller property of the invariant Gibbs dynamics for the cubic SNLW on T 2 with a slightly smoothed noise. The ergodic property of the hyperbolic Φ 3 3 -model is a challenging problem, in particular due to its non-defocusing nature.…”

mentioning

confidence: 99%

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

Oh¹,

Okamoto²,

Tolomeo³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

mentioning

confidence: 99%

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

Oh¹,

Okamoto²,

Tolomeo³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We thus believe that its proof itself is of interest. Finally we remark that the uniqueness of the invariant measures was investigated in [31] for the case of d = 1 with space-time white noise, and in [7] in the case d = 2 for a slightly more regular noise than space-time white noise. The case d = 2 with space-time white noise is open, but we expect the uniqueness of the Gibbs measure.…”

Section: Introductionmentioning

confidence: 99%

Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein–Gordon equation

2022

View full text Add to dashboard Cite

We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein–Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.

show abstract

“…We thus believe that its proof itself is of interest. Finally we remark that the uniqueness of the invariant measures is investigated in [21] for the case of d = 1 with space-time white noise, and in [5] in the case d < 2 for a slightly more regular noise than space-time white noise.…”

Section: Introductionmentioning

confidence: 99%

Non relativistic and ultra relativistic limits in 2d stochastic nonlinear damped Klein-Gordon equation

Fukuizumi,

Hoshino,

Inui

2021

Preprint

View full text Add to dashboard Cite

We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.

show abstract

On the unique ergodicity for a class of 2 dimensional stochastic wave equations

Cited by 3 publications

References 19 publications

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

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

Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein–Gordon equation

Non relativistic and ultra relativistic limits in 2d stochastic nonlinear damped Klein-Gordon equation

Contact Info

Product

Resources

About