On the two-dimensional hyperbolic stochastic sine-Gordon equation

Tadahiro Oh; Tristan Robert; Philippe Sosoe; Yuzhao Wang

doi:10.1007/s40072-020-00165-8

On the two-dimensional hyperbolic stochastic sine-Gordon equation

Tadahiro Oh, Tristan Robert, Philippe Sosoe, Yuzhao Wang

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

217 Downloads (Pure)

Abstract

We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary Gaussian multiplicative chaos, we prove local well-posedness of SSG for any value of a parameter β2>0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (Commun Math Phys 341(3):933–989, 2016) and Chandra et al. (The dynamical sine-Gordon model in the full subcritical regime, arXiv:1808.02594 [math.PR], 2018), where the parameter is restricted to the subcritical range: 0<β2<8π. We also present a triviality result for the unrenormalized SSG.

Original language	English
Pages (from-to)	1–32
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Volume	9
Issue number	1
Early online date	5 Feb 2020
DOIs	https://doi.org/10.1007/s40072-020-00165-8
Publication status	Published - Mar 2021

Bibliographical note

Funding Information:
T.O. and T.R. were supported by the European Research Council (Grant No. 637995 “ProbDynDispEq”). P.S. was partially supported by NSF Grant DMS-1811093. The authors would like to thank the anonymous referees for the helpful comments.

Publisher Copyright:
© 2020, The Author(s).

Keywords

Gaussian multiplicative chaos
Renormalization
Sine-Gordon equation
Stochastic sine-Gordon equation
White noise

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

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Cite this

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title = "On the two-dimensional hyperbolic stochastic sine-Gordon equation",

abstract = "We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary Gaussian multiplicative chaos, we prove local well-posedness of SSG for any value of a parameter β2>0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (Commun Math Phys 341(3):933–989, 2016) and Chandra et al. (The dynamical sine-Gordon model in the full subcritical regime, arXiv:1808.02594 [math.PR], 2018), where the parameter is restricted to the subcritical range: 0<β2<8π. We also present a triviality result for the unrenormalized SSG.",

keywords = "Gaussian multiplicative chaos, Renormalization, Sine-Gordon equation, Stochastic sine-Gordon equation, White noise",

author = "Tadahiro Oh and Tristan Robert and Philippe Sosoe and Yuzhao Wang",

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AU - Oh, Tadahiro

AU - Robert, Tristan

AU - Sosoe, Philippe

AU - Wang, Yuzhao

N1 - Funding Information: T.O. and T.R. were supported by the European Research Council (Grant No. 637995 “ProbDynDispEq”). P.S. was partially supported by NSF Grant DMS-1811093. The authors would like to thank the anonymous referees for the helpful comments. Publisher Copyright: © 2020, The Author(s).

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N2 - We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary Gaussian multiplicative chaos, we prove local well-posedness of SSG for any value of a parameter β2>0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (Commun Math Phys 341(3):933–989, 2016) and Chandra et al. (The dynamical sine-Gordon model in the full subcritical regime, arXiv:1808.02594 [math.PR], 2018), where the parameter is restricted to the subcritical range: 0<β2<8π. We also present a triviality result for the unrenormalized SSG.

AB - We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary Gaussian multiplicative chaos, we prove local well-posedness of SSG for any value of a parameter β2>0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (Commun Math Phys 341(3):933–989, 2016) and Chandra et al. (The dynamical sine-Gordon model in the full subcritical regime, arXiv:1808.02594 [math.PR], 2018), where the parameter is restricted to the subcritical range: 0<β2<8π. We also present a triviality result for the unrenormalized SSG.

KW - Gaussian multiplicative chaos

KW - Renormalization

KW - Sine-Gordon equation

KW - Stochastic sine-Gordon equation

KW - White noise

UR - https://arxiv.org/abs/1907.06055

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EP - 32

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JF - Stochastics and Partial Differential Equations: Analysis and Computations

IS - 1

ER -

On the two-dimensional hyperbolic stochastic sine-Gordon equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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Cite this