Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

Justin Forlano; Leonardo Tolomeo

doi:10.1007/s40072-025-00407-7

Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

Justin Forlano, Leonardo Tolomeo^*

^*Corresponding author for this work

School of Mathematics

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

Abstract

We consider the stochastic damped nonlinear wave equation ∂t2u+∂tu+u-Δu+u3=2⟨∇⟩-sξ on the two-dimensional torus T2, where ξ denotes a space-time white noise and s>0. We show that the measure μ→s corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.

Original language	English
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Early online date	9 Dec 2025
DOIs	https://doi.org/10.1007/s40072-025-00407-7
Publication status	E-pub ahead of print - 9 Dec 2025

Keywords / Materials (for Non-textual outputs)

Absolute continuity
Gaussian measure
Invariant measure
Quasi-invariance
Stochastic nonlinear wave equation
White noise

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10.1007/s40072-025-00407-7Licence: Creative Commons: Attribution (CC-BY)

https://arxiv.org/abs/2409.20451

Cite this

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title = "Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation",

abstract = "We consider the stochastic damped nonlinear wave equation ∂t2u+∂tu+u-Δu+u3=2⟨∇⟩-sξ on the two-dimensional torus T2, where ξ denotes a space-time white noise and s>0. We show that the measure μ→s corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.",

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AU - Forlano, Justin

AU - Tolomeo, Leonardo

N1 - Publisher Copyright: © The Author(s) 2025.

PY - 2025/12/9

Y1 - 2025/12/9

N2 - We consider the stochastic damped nonlinear wave equation ∂t2u+∂tu+u-Δu+u3=2⟨∇⟩-sξ on the two-dimensional torus T2, where ξ denotes a space-time white noise and s>0. We show that the measure μ→s corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.

AB - We consider the stochastic damped nonlinear wave equation ∂t2u+∂tu+u-Δu+u3=2⟨∇⟩-sξ on the two-dimensional torus T2, where ξ denotes a space-time white noise and s>0. We show that the measure μ→s corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.

KW - Absolute continuity

KW - Gaussian measure

KW - Invariant measure

KW - Quasi-invariance

KW - Stochastic nonlinear wave equation

KW - White noise

UR - https://www.scopus.com/pages/publications/105024896147

U2 - 10.1007/s40072-025-00407-7

DO - 10.1007/s40072-025-00407-7

M3 - Article

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SN - 2194-0401

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

ER -

Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

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SingStocDispDyn: Singular Stochastic Dispersive Dynamics

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Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

Abstract

Keywords / Materials (for Non-textual outputs)

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SingStocDispDyn: Singular Stochastic Dispersive Dynamics

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