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<p>We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato–Debussche regime. The approach is essentially algebraic and involves using the generator equation of the SPDE at stationarity. Our results in particular cover the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Phi ^4_\delta \)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Φ</mi> <mi>δ</mi> <mn>4</mn> </msubsup> </math></EquationSource> </InlineEquation> measures in dimensions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\delta &lt;\frac{14}{5}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>&lt;</mo> <mfrac> <mn>14</mn> <mn>5</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.</p>

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Non-Gaussianity of invariant measures to SPDEs in Da Prato–Debussche regime

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月19日

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  • Ajay Chandra,
  • Ilya Chevyrev

摘要

We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato–Debussche regime. The approach is essentially algebraic and involves using the generator equation of the SPDE at stationarity. Our results in particular cover the \(\Phi ^4_\delta \) Φ δ 4 measures in dimensions \(\delta <\frac{14}{5}\) δ < 14 5 , which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.