Recentering with Malliavin derivative

1. Problem Statement

The paper addresses a fundamental challenge in the theory of Regularity Structures (introduced by M. Hairer), which is used to solve singular Stochastic Partial Differential Equations (SPDEs). Specifically, it focuses on the convergence proofs of renormalised models.

• The Core Difficulty: Proving the convergence of the renormalised model typically requires handling two simultaneous analytical procedures:
  1. Renormalisation: Removing divergences (often via the BPHZ scheme).
  2. Recentering: Adjusting the model so that it behaves correctly under changes of the expansion base point (from $x$ to $y$).
• The Gap in Literature: Different approaches exist in the literature to prove convergence using spectral gap inequalities (e.g., works by Hairer, Labbé, Otto, etc.).
  • Some use decorated trees (Hairer's original framework).
  • Others use multi-indices (Labbé, Otto, etc.).
  • A specific algebraic map, denoted as $d\Gamma$, is central to the "recentering" step in these proofs.
  • The Issue: It was unclear whether the $d\Gamma$ map defined in the decorated tree framework (specifically in Bruned et al. [9]) was algebraically equivalent to the maps used in the multi-index framework (Labbé [18]) or other spectral gap proofs (e.g., [15], [3]). The lack of a unified algebraic understanding made it difficult to compare and unify these convergence proofs.

2. Methodology

The authors employ a rigorous algebraic approach based on Hopf algebras of decorated trees to unify these different frameworks.

• Framework: They utilize the algebraic structures of decorated trees ($\mathcal{T}$) and forests, extending the definitions found in [9] to accommodate Malliavin derivatives.
• Key Algebraic Tools:
  • Decorated Trees: They define trees with node and edge decorations, distinguishing between the noise $\Xi$ and its infinitesimal perturbation $\dot{\Xi}$ (representing the Malliavin derivative $\delta\xi$).
  • Degrees: They introduce three distinct degree maps ($\deg_0, \deg_1, \deg_2$) to track regularity, specifically accounting for the gain in regularity when a Malliavin derivative is applied.
  • Preparation Maps ($R$): They define renormalisation maps that commute with the abstract Malliavin derivative $D_\Xi$.
  • Coproducts and Co-actions: They utilize co-actions $\Delta_i$ and coproducts $\Delta^+_i$ to define the re-expansion maps $\Gamma_{xy}$ and the central recentering map $d\Gamma$.
• Strategy:
  1. Define the recentering map $d\Gamma$ explicitly on decorated trees using Hopf algebraic operations (involving $\Gamma_{yx}$, characters $f_x$, and projections $Q_0$).
  2. Prove that this map satisfies two specific characterisations previously used in different literature:
     • A derivative-based characterisation (relating to the Malliavin derivative of the model).
     • A recursive characterisation (relating to the algebraic structure of modelled distributions).

3. Key Contributions

A. Unification of Spectral Gap Proofs

The primary contribution is Theorem 1.1, which states that the algebraic parts of convergence proofs via spectral gap inequalities in various frameworks ([17], [15], [3]) all rely on the same recentering map $d\Gamma$, despite being presented differently. The authors provide the explicit expression of this map on decorated trees (Equation 1.3 / 2.23).

B. Two Equivalent Characterisations of $d\Gamma$

The paper proves that the map $d\Gamma$ defined in the decorated tree framework is equivalent to the maps used in other approaches through two main theorems:

1. Derivative Approach (Theorem 3.2):

   • The authors prove that $d\Gamma$ satisfies the identity:
     $$\partial^n (\Pi_y d\Gamma_{yx} \tau - \delta \Pi_x \tau)(y) = 0$$
     for all $n$ within a certain regularity bound.
   • This confirms that $d\Gamma$ correctly reconstructs the Malliavin derivative of the model ($\delta \Pi_x$), matching the characterisation used in the multi-index approach (Labbé [18]).

2. Recursive Definition (Theorem 3.4):

   • The authors show that a variant of the map, $d\Gamma_2$, satisfies a specific recursive induction formula:
     • $d\Gamma_{yx}(\sigma \tau) = d\Gamma_{yx}\tau \times \Gamma_{yx}\sigma + \Gamma_{yx}\tau \times d\Gamma_{yx}\sigma$
     • $d\Gamma_{yx}I_a(\tau) = (I_a + J_a(y))(d\Gamma_{yx}\tau) - \Gamma_{xy} P (J_a(x)(d\Gamma_{xx}\tau))$
   • This matches the recursive definition of the algebraic map $\tilde{f}^\tau_x$ used in the work of Chandra and Otto ([15]), thereby bridging the gap between the tree-based and recursive approaches.

C. Technical Refinements

• Assumption 1 & 2: The paper introduces specific assumptions regarding the degrees of trees containing Malliavin derivatives. These assumptions ensure that trees with a "planted" Malliavin derivative have positive degree, which is crucial for the reconstruction argument to hold.
• Dual Side Equivalence: Proposition 3.3 establishes the equivalence between the tree-based definition and the dual definition used in multi-index frameworks, proving that the results are consistent regardless of the representation (trees vs. multi-indices).

4. Results

• Explicit Formula: The paper provides a concrete, Hopf-algebraic formula for the recentering map $d\Gamma$ (Equations 2.23 and 2.24) that works for a broad class of singular SPDEs.
• Verification: The authors verify that this formula yields the correct Malliavin derivative of the model when reconstructed, satisfying the necessary spectral gap conditions.
• Consistency: The work demonstrates that the "dictionary" between different combinatorial frameworks (trees vs. multi-indices) is consistent at the level of the recentering map, resolving previous ambiguities about whether the maps in [9], [18], and [15] were truly the same.

5. Significance

• Theoretical Unification: This paper provides a "Rosetta Stone" for the algebraic side of Regularity Structures. It proves that the various spectral gap proofs found in the literature are not just similar in spirit but are algebraically identical in their core mechanism.
• Simplification of Future Proofs: By establishing a unified definition of $d\Gamma$, future researchers can switch between the decorated tree formalism and the multi-index formalism without worrying about algebraic inconsistencies in the recentering step.
• Robustness of Renormalisation: The work reinforces the robustness of the BPHZ renormalisation scheme when combined with Malliavin calculus, showing that the interplay between renormalisation and recentering can be handled systematically via Hopf algebraic structures.
• Foundation for Higher Order: The introduction of higher-order derivative characterisations (Theorem 3.2) extends the applicability of these proofs to more complex equations where higher-order regularity is required.

In summary, Bruned and Minguella have successfully demystified the algebraic structure underlying the convergence of renormalised models, proving that the recentering maps used in disparate approaches are, in fact, variations of a single, unified algebraic object.