<p>We consider the stochastic damped nonlinear wave equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\partial _t^{2}u+\partial _t u+u-\Delta u +u^{3} = \sqrt{2} {\langle {\nabla }\rangle ^{-s}} \xi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>∂</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mi>u</mi> <mo>+</mo> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msup> <mi>u</mi> <mn>3</mn> </msup> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <msup> <mrow> <mo stretchy="false">⟨</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">⟩</mo> </mrow> <mrow> <mo>-</mo> <mi>s</mi> </mrow> </msup> <mi>ξ</mi> </mrow> </math></EquationSource> </InlineEquation> on the two-dimensional torus <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb T^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">T</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> denotes a space-time white noise and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that the measure <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\vec {\mu }_s$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>μ</mi> <mo stretchy="false">→</mo> </mover> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.</p>

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

期刊论文
出版模式：

开放获取
发表日期：

2025年12月9日

查看全文

Justin Forlano,
Leonardo Tolomeo

摘要

We consider the stochastic damped nonlinear wave equation $\partial _t^{2}u+\partial _t u+u-\Delta u +u^{3} = \sqrt{2} {\langle {\nabla }\rangle ^{-s}} \xi $ $\partial_{t}^{2} u + \partial_{t} u + u - Δ u + u^{3} = \sqrt{2} {⟨ \nabla ⟩}^{- s} ξ$ on the two-dimensional torus $\mathbb T^2$ $T^{2}$ , where $\xi $ $ξ$ denotes a space-time white noise and $s>0$ $s > 0$ . We show that the measure $\vec {\mu }_s$ ${\vec{μ}}_{s}$ corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.