Bogoliubov type recursions for renormalisation in regularity structures

This paper reformulates the renormalisation framework for Hairer's regularity structures by introducing Bogoliubov-type recursions, analogous to the Connes-Kreimer approach, to clarify the interplay between positive and negative renormalisation and apply it to singular stochastic partial differential equations.

The Gist

The Big Picture: Taming the Wild Storm

Imagine you are trying to predict the weather, but the atmosphere is so chaotic that the equations you use to describe it break down. The numbers you get are infinite or nonsensical. In the world of physics and mathematics, this happens with Singular Stochastic Partial Differential Equations (SPDEs). These are equations used to model things like heat spreading through a material that is being shaken by random, violent noise (like a storm).

For a long time, mathematicians couldn't solve these equations because the "noise" was too rough. Then, a mathematician named Martin Hairer invented a new framework called Regularity Structures. Think of this as a new kind of telescope that allows you to see the fine details of the chaos and make sense of it.

However, using this telescope requires a very specific, complex cleaning process called renormalisation. This paper by Yvain Bruned and Kurusch Ebrahimi-Fard is about making that cleaning process clearer, more systematic, and easier to understand.

The Core Problem: Two Types of Mess

To solve these equations, you have to deal with two different kinds of "mess":

1. The "Recentering" Mess (Positive Renormalisation): Imagine you are trying to describe a landscape, but your map is shifted. You need to shift your map back so that "zero" is actually at the point you are standing. In math, this means re-centering polynomials so they match the local reality.
2. The "Infinite Noise" Mess (Negative Renormalisation): This is the big one. When you multiply the random noise by itself, you get infinity. You need a way to subtract these infinities so you are left with a finite, usable number.

The paper argues that these two messy problems are actually two sides of the same coin, and they can be solved using a specific mathematical recipe.

The Analogy: The "Bogoliubov" Recipe

The authors introduce a method called Bogoliubov-type recursions. To understand this, imagine you are a chef trying to make a perfect soup, but your ingredients are contaminated with sand (the infinities).

1. The Ingredients (Decorated Trees): In this math world, the ingredients are represented by trees. These aren't real trees, but diagrams with branches and leaves. Each branch has a label (a decoration) telling you what kind of "ingredient" it is.
2. The Recipe (The Recursion): You can't just throw the whole tree into the pot. You have to break it down. The "recursion" is a step-by-step instruction manual:
   • Look at a small branch.
   • Check if it has sand (divergence).
   • If it does, use a special tool to scrape the sand off (this is the counterterm).
   • Put the clean branch back together.
   • Repeat this process for every branch, working from the smallest twigs up to the main trunk.

The paper shows that this "scraping" process follows a very elegant pattern, similar to a recipe used in quantum physics (the BPHZ method), but adapted for these specific "tree" diagrams.

The Magic Tool: The "Birkhoff" Split

The paper relies on a concept called Algebraic Birkhoff Factorisation.

Imagine you have a tangled ball of yarn (the messy equation). You want to separate it into two distinct balls:

• Ball A (The Clean Part): This is the useful, finite part of the solution.
• Ball B (The Trash): This is the infinite garbage you need to throw away.

The authors show that there is a mathematical "magic trick" (a decomposition) that guarantees you can always separate the yarn into these two perfect balls, provided you follow their specific recursion rules. They prove that this trick works even when the "trees" are complicated and not perfectly connected, which was a major hurdle in previous attempts.

The Two Main Applications

The paper applies this new, clearer recipe to the two types of renormalisation mentioned earlier:

1. Positive Renormalisation (The Map Shift): They show how to use their recursion to perfectly re-center the polynomials. It's like realizing your map was drawn from the wrong city center, and using their formula to instantly shift the "zero point" to where you actually are, without messing up the rest of the map.
2. Negative Renormalisation (The Sand Removal): They apply the same logic to remove the infinities. They treat the "trash" (the infinities) as a specific type of algebraic object that can be systematically identified and subtracted, leaving behind a clean, solvable equation.

Why This Matters (According to the Paper)

Before this paper, the connection between the "tree" diagrams used in Hairer's theory and the famous "Bogoliubov" recursion used in quantum physics was a bit fuzzy. It was like knowing two different chefs were making the same dish but using different, confusing terminology.

This paper acts as a translator. It says: "Look, the way you clean up these SPDEs is actually the exact same mathematical structure as the way you clean up quantum physics problems."

By defining these recursions clearly, the authors provide a new, robust toolkit. They prove that the "cleaning" process (renormalisation) is not just a hack, but a rigorous, logical process that can be broken down into simple, repeatable steps. This makes the theory of Regularity Structures more solid and easier for other mathematicians to use and build upon.

Summary in One Sentence

This paper takes a complex mathematical method for solving chaotic equations, breaks it down into a step-by-step "recipe" using tree diagrams, and proves that this recipe is a universal tool for cleaning up both the "shifted maps" and the "infinite noise" in these equations.

Technical Summary

Technical Summary: Bogoliubov-type recursions for renormalisation in regularity structures

Problem Statement
The theory of Regularity Structures (RS), introduced by Hairer, provides a robust framework for solving singular Stochastic Partial Differential Equations (SPDEs). A critical component of this theory is the renormalisation procedure, which addresses two distinct issues: the recentering of distributions around a point (Positive Renormalisation) and the removal of divergences arising from ill-defined distributional products (Negative Renormalisation). Algebraically, these procedures are underpinned by cointeracting bialgebras and an algebraic Birkhoff decomposition of bialgebra morphisms. However, the specific link between the renormalisation of SPDEs and the classical Bogoliubov recursions used in perturbative Quantum Field Theory (QFT) via the BPHZ method has remained somewhat implicit. The paper aims to clarify this link, specifically addressing the fact that the Hopf algebras involved in RS are not necessarily connected, unlike the standard Connes-Kreimer Hopf algebra of rooted trees used in QFT.

Methodology
The authors revisit the algebraic foundations of renormalisation by adapting the Connes-Kreimer formulation of BPHZ renormalisation to the specific context of Regularity Structures. The methodology proceeds through the following steps:

1. Algebraic Birkhoff Factorisation: The paper first recalls the standard algebraic Birkhoff decomposition for connected graded Hopf algebras, where characters are decomposed into counterterm and renormalised parts using a Rota-Baxter projection. It then extends this to a setting involving a right-comodule over a Hopf algebra, replacing the coproduct with a coaction to accommodate the specific structures of RS.
2. Decorated Trees and Comodule-Hopf Structures: The authors define a space of decorated trees ($RT_D$) representing the combinatorial objects in RS. They establish a comodule-Hopf algebra structure on these trees. A key technical challenge is that the relevant Hopf algebra is not connected. To circumvent this, the authors define a modified reduced coproduct ($\Delta^+_{red}$) and a modified reduced coaction ($\hat{\Delta}^+$).
3. Bogoliubov-type Recursions: Central to the approach is the introduction of a family of Taylor-jet operators ($T_{\alpha, x, y}$) which act as Rota-Baxter maps. These operators facilitate the splitting of the target function space into polynomial and non-polynomial parts. Using these operators and the modified reduced coproduct, the authors define a recursive scheme (Definition 3.10) that mirrors Bogoliubov's preparation map. This recursion generates a counterterm character ($\phi^-$) and a renormalised character ($\phi^+$).
4. Application to SPDEs: The framework is applied to the two specific renormalisation procedures in RS:
   • Positive Renormalisation: This involves the construction of the recentering map (the model $\Pi_x$). The authors show that this map can be derived via the Bogoliubov recursion, linking the twisted antipode $\tilde{A}^+$ to the counterterm character.
   • Negative Renormalisation: This addresses the removal of divergences. The authors utilize a coaction on a space of forests (quotiented by positive-degree trees) and a Rota-Baxter map based on expectation values to define the negative renormalisation, recovering the BPHZ algorithm within the RS framework.

Key Contributions and Results

• New Birkhoff Factorisation: The paper establishes a new Birkhoff factorisation distinct from the original Connes-Kreimer formulation. This is achieved by handling non-connected Hopf algebras through a modified reduced coproduct and a comodule structure.
• Bogoliubov Recursion for RS: The authors explicitly define Bogoliubov-type recursions for the counterterms in Regularity Structures (Theorem 3.13). They prove that the counterterm map $\phi^-_{x, \bar{x}}$ is an algebra morphism from the positive part of the tree space to the space of polynomial functions, and the renormalised map $\phi^+_{x, \bar{x}}$ is an algebra morphism to the full function space.
• Connection to Twisted Antipodes: A significant result (Proposition 3.15) demonstrates that the counterterm character constructed via the Bogoliubov recursion is equivalent to the composition of the original character with the "twisted antipode" ($\tilde{A}^+$) previously defined in the RS literature. This bridges the gap between the recursive definition of the model and the algebraic Hopf algebraic formulation.
• Invariance Properties: Under specific assumptions regarding the family of characters (Assumption 2), the authors prove that the renormalised character and the preparation map are invariant with respect to the recentering parameter $\bar{x}$ (Theorem 3.17). This confirms that the resulting model does not depend on the arbitrary choice of polynomial recentering.
• Unified View of Renormalisation: The paper unifies the treatment of Positive and Negative renormalisation. Positive renormalisation is shown to rely on the comodule structure and the modified coproduct, while Negative renormalisation is shown to be a direct application of the algebraic Birkhoff decomposition on a connected Hopf algebra of forests, utilizing expectation values as the Rota-Baxter projection.

Significance
The paper claims to make the link between the renormalisation procedures in Regularity Structures and the algebraic BPHZ renormalisation in QFT more precise. By formulating the recentering map and the removal of divergences as solutions to factorisation problems involving Bogoliubov-type recursions, the authors provide a clearer algebraic understanding of Hairer's theory.

The work offers new tools for the analysis of SPDEs, particularly in the context of local error analysis for numerical schemes, where similar algebraic structures (Rota-Baxter maps and Birkhoff factorisations) are relevant. The authors note that while the Connes-Kreimer approach relies on connected Hopf algebras, the specific context of SPDEs requires the more general comodule-Hopf algebra framework developed here. This formulation allows for a rigorous definition of the "model" (a critical component of RS) as an alternative to previous constructions, ensuring that the recentering map is independent of a priori polynomial choices. The paper does not propose new experimental applications but rather refines the theoretical machinery underpinning the solution theory of singular SPDEs.