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<p>We present an example of a linear partial differential equation whose Cauchy problem becomes well-posed when perturbed by noise. Specifically, we make clear how a suitable multiplicative Stratonovich perturbation of Brownian type renders a weakly hyperbolic operator with double involutive characteristics well-posed in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-category, while its deterministic counterpart is only well-posed in the Gevrey <i>s</i> classes with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( 1 \le s &lt;2 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>s</mi> <mo>&lt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Regularization by noise for Gevrey well-posedeness of a weakly hyperbolic operator

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月23日

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  • Enrico Bernardi,
  • Alberto Lanconelli

摘要

We present an example of a linear partial differential equation whose Cauchy problem becomes well-posed when perturbed by noise. Specifically, we make clear how a suitable multiplicative Stratonovich perturbation of Brownian type renders a weakly hyperbolic operator with double involutive characteristics well-posed in the \(C^{\infty }\) C ∞ -category, while its deterministic counterpart is only well-posed in the Gevrey s classes with \( 1 \le s <2 \) 1 ≤ s < 2 .