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<p>Malliavin calculus provides a characterization of the centered model in regularity structures that is stable under removing the small-scale cut-off. In conjunction with a spectral gap inequality, it yields the stochastic estimates of the model. This becomes transparent on the level of a notion of model that parameterizes the solution manifold, and thus is indexed by multi-indices rather than trees, and which allows for a more geometric than combinatorial perspective. In these lecture notes, this is carried out for a PDE with heat operator, a cubic nonlinearity, and driven by additive noise, reminiscent of the stochastic quantization of the Euclidean <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\phi ^4\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ϕ</mi> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> model. More precisely, we informally motivate our notion of the model <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\Pi ,\Gamma )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Π</mi> <mo>,</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as charts and transition maps, respectively, of the nonlinear solution manifold. These geometric objects are algebrized in terms of formal power series, and their algebra automorphisms. We will assimilate the directional Malliavin derivative to a tangent vector of the solution manifold. This means that it can be treated as a modelled distribution, thereby connecting stochastic model estimates to pathwise solution theory, with its analytic tools of reconstruction and integration. We unroll an inductive calculus that in an automated way applies to the full subcritical range.</p>

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Lecture notes on Malliavin calculus in regularity structures

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月9日

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  • Lucas Broux,
  • Felix Otto,
  • Markus Tempelmayr

摘要

Malliavin calculus provides a characterization of the centered model in regularity structures that is stable under removing the small-scale cut-off. In conjunction with a spectral gap inequality, it yields the stochastic estimates of the model. This becomes transparent on the level of a notion of model that parameterizes the solution manifold, and thus is indexed by multi-indices rather than trees, and which allows for a more geometric than combinatorial perspective. In these lecture notes, this is carried out for a PDE with heat operator, a cubic nonlinearity, and driven by additive noise, reminiscent of the stochastic quantization of the Euclidean \(\phi ^4\) ϕ 4 model. More precisely, we informally motivate our notion of the model \((\Pi ,\Gamma )\) ( Π , Γ ) as charts and transition maps, respectively, of the nonlinear solution manifold. These geometric objects are algebrized in terms of formal power series, and their algebra automorphisms. We will assimilate the directional Malliavin derivative to a tangent vector of the solution manifold. This means that it can be treated as a modelled distribution, thereby connecting stochastic model estimates to pathwise solution theory, with its analytic tools of reconstruction and integration. We unroll an inductive calculus that in an automated way applies to the full subcritical range.