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Lecture notes on Malliavin calculus in regularity structures

Lucas Broux, Felix Otto, Markus Tempelmayr

TL;DR

The work develops a Malliavin-calculus–driven framework within regularity structures to study a subcritical singular SPDE with a cubic nonlinearity driven by rough noise. By parameterizing the nonlinear solution manifold with multi-indices and constructing a centered model $(Π,Γ)$, the authors establish a robust tangent-space theory via a Malliavin derivative map $\mathrm{d}\Gamma^*$ and prove uniform-in-$\rho$ estimates as mollification vanishes, after a BPHZ renormalization that yields divergent constants $c_k$. The analysis hinges on a precise algebraic–analytic induction that uses reconstruction, Besov-type norms, and a spectral-gap inequality to control fluctuations through the carré-du-champs; a robust relation between $\deltaΠ$ and $\deltaΠ^-$ replaces non-robust renormalization effects. The change-of-base-point machinery, particularly the action of $\Gamma_{xx'}^*$ and its triangularity properties, ties local charts into a global model, enabling pathwise reconstruction and tangent-space modeling that connect stochastic estimates to deterministic solution theory. Overall, the results provide a coherent, scalable method for obtaining uniform stochastic estimates in the subcritical regime and clarify how Malliavin calculus can be integrated with regularity-structure reconstructions to study singular SPDEs.

Abstract

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 $φ^4$ model. More precisely, we informally motivate our notion of the model $(Π,Γ)$ 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 regime.

Lecture notes on Malliavin calculus in regularity structures

TL;DR

The work develops a Malliavin-calculus–driven framework within regularity structures to study a subcritical singular SPDE with a cubic nonlinearity driven by rough noise. By parameterizing the nonlinear solution manifold with multi-indices and constructing a centered model , the authors establish a robust tangent-space theory via a Malliavin derivative map and prove uniform-in- estimates as mollification vanishes, after a BPHZ renormalization that yields divergent constants . The analysis hinges on a precise algebraic–analytic induction that uses reconstruction, Besov-type norms, and a spectral-gap inequality to control fluctuations through the carré-du-champs; a robust relation between and replaces non-robust renormalization effects. The change-of-base-point machinery, particularly the action of and its triangularity properties, ties local charts into a global model, enabling pathwise reconstruction and tangent-space modeling that connect stochastic estimates to deterministic solution theory. Overall, the results provide a coherent, scalable method for obtaining uniform stochastic estimates in the subcritical regime and clarify how Malliavin calculus can be integrated with regularity-structure reconstructions to study singular SPDEs.

Abstract

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 model. More precisely, we informally motivate our notion of the model 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 regime.
Paper Structure (49 sections, 1 theorem, 328 equations, 3 figures, 1 table)

This paper contains 49 sections, 1 theorem, 328 equations, 3 figures, 1 table.

Table of Contents

  1. Motivation and setting
  2. A nonlinear partial differential equation with rough right-hand side
  3. Structure of these lecture notes
  4. Bibliographical context
  5. A random noise with symmetries in law and restriction to the singular and subcritical range
  6. Renormalization through a counterterm , multi-indices, and the homogeneity
  7. Parameterization of the solution manifold
  8. Power series representation of the parameterization
  9. Characterization of and as formal power series
  10. Noise homogeneity and population conditions
  11. Scale invariance of the solution manifold and homogeneity revisited
  12. Uniqueness of Pi given Pi-
  13. Uniqueness of c via BPHZ-choice of renormalization
  14. Shift-invariance of the solution manifold, general base-points, and corresponding centered model
  15. The change-of-base-point transformation
  16. ...and 34 more sections

Key Result

Theorem 1

Suppose that the centered Gaussian ensemble on Schwartz distributions on $\mathbb{R}^{1+d}$ satisfies the symmetries ck19 -- ck21, and satisfies ck81bis for some $s$ with ck53 and ck28. Moreover, we assume $D \geq 3$ and Then we have The implicit multiplicative constants only depend on $d$, $s$, $\beta$, $\gamma$, $p$, and on $\psi$ only through the semi-norms, but not on $x$, $r$, and $\rho$.

Figures (3)

  • Figure 1: Heuristic visualization of the parameterization. When $\lambda = 0$ the solution manifold is affine and parameterized by analytic functions $p$. When $\lambda \neq 0$ we expect it to be still parameterized by $p$ in a non-linear way.
  • Figure 2: Heuristic visualization of the change-of-base-point transformation $\Gamma$. Informally, $\Pi_x$ and $\Pi_y$ act as (inverse) "charts" on the solution manifold of the $\phi$'s, while $\Gamma_{y x}$ acts as a "transition function" between these two "charts".
  • Figure 3: Heuristic visualization of the tangent space to the solution manifold.

Theorems & Definitions (34)

  • proof : Proof of \ref{['ao08']} & \ref{['temp17']}
  • proof : Proof of \ref{['ck33']}
  • proof : Proof of uniqueness for $\Pi_\beta$
  • proof : Proof that \ref{['ck57']} & \ref{['ck52ter']} fix $c$
  • proof : Proof of \ref{['ck62']}
  • proof : Proof of \ref{['ap04']}
  • proof : Proof of \ref{['cw46']} & \ref{['cw46ter']} & \ref{['cw46bis']}
  • proof : Proof of \ref{['cw44']}
  • Theorem 1
  • Remark 1
  • ...and 24 more