[Paper Review] Random models on regularity-integrability structures

I. Bailleul, Masato Hoshino|arXiv (Cornell University)|Oct 16, 2023

Stochastic processes and financial applications4 citations

TL;DR

This paper introduces the concept of a regularity-integrability structure to unify the analysis of random models and their Malliavin derivatives in the context of singular stochastic PDEs. It establishes a convergence result for a broad class of renormalisation procedures—including BPHZ—by leveraging spectral gap assumptions and an inductive control of stochastic objects, extending prior results in a more general and streamlined framework.

ABSTRACT

We prove a convergence result for a large class of random models that encompasses the case of the BPHZ models used in the study of singular stochastic PDEs. We introduce for that purpose a useful variation on the notion of regularity structure called a regularity-integrability structure. It allows to deal in a single elementary setting with models on a usual regularity structure and their first order Malliavin derivative.

Motivation & Objective

• To generalize the convergence theory of random models in singular stochastic PDEs beyond the scope of existing BPHZ and Otto-type frameworks.
• To unify the treatment of a model and its first-order Malliavin derivative within a single analytical structure, enabling joint control of stochastic objects.
• To provide a new, simplified proof of convergence for renormalised models that avoids intricate algebraic machinery while maintaining broad applicability.
• To extend the convergence result of Hairer & Steele [20] to a larger class of renormalisation procedures, including BPHZ, using a novel regularity-integrability structure.
• To establish robust estimates for the lifted operator $Σ^{‘M}$ acting on modelled distributions, ensuring stability under model perturbations.

Proposed method

• Introduces a new algebraic-analytic framework called a regularity-integrability structure, which extends standard regularity structures by incorporating Malliavin-type derivatives.
• Defines a compatible pair of models $(‘M, ̅‘M)$ on a base and extended structure, ensuring consistency under the action of an abstract integration map $σ$.
• Constructs a lifted operator $Σ^{‘M}$ that combines the abstract integration map $σ$, a correction term $σ^{‘M}(x)$, and a reconstruction-based term $Σ^{‘M}(x;f,\Lambda)$.
• Employs a spectral gap assumption on the noise law to control $L^p$ norms of modelled quantities via their Gâteaux derivatives, enabling induction-based estimates.
• Uses an inductive argument to propagate control of $L^p$ norms from $λ = 1$ to all $0 < λ \leq 1$, ensuring uniform bounds across scales.
• Derives quantitative stability estimates via the norm $|“|“ \cdot |“|“_{{\bf c};w_{2c+b}}$, showing continuity of the lifted operator under model and function perturbations.

Experimental results

Research questions

• RQ1 Can the convergence of random models in singular SPDEs be established in a framework that unifies the model and its Malliavin derivative?
• RQ2 How can the spectral gap assumption be leveraged to replace the complex cumulant-based analysis of Chandra & Hairer [12]?
• RQ3 What structural extension of regularity structures allows for a unified treatment of renormalisation procedures beyond BPHZ?
• RQ4 Can the convergence proof be generalized to a broader class of renormalisation rules while preserving the same quantitative estimates?
• RQ5 What is the precise stability behavior of the lifted operator $Σ^{‘M}$ under perturbations of the model and the modelled distribution?

Key findings

• The paper establishes that the lifted operator $Σ^{‘M}f$ belongs to the space $D^{(γ+\beta_0,r)}(\overline{\Gamma})_{w_{2c+b}}$ under appropriate regularity and integrability conditions.
• It proves the estimate $\llparenthesis\mathcal{K}^{‘M}f\rrparenthesis_{(\gamma+\beta_0,r);w_{2c+b}} \lesssim (1+\|\Pi\|_{{\bf c};w_c})(1+\|\Gamma\|_{{\bf c};w_c})\llparenthesis f\rrparenthesis_{{\bf c};w_b} + \llbracket\Lambda\rrbracket_{{\bf c};w_{c+b}}^{\Pi,f}$, showing control of the lifted function in terms of the input model and reconstruction.
• It derives a stability estimate: $|“|“ \mathcal{K}^{‘M_1}f_1; \mathcal{K}^{‘M_2}f_2 |“|“_{(\gamma+\beta_0,r);w_{2c+b}} \leq C_\lambda (|“|“ \u2018M_1;\u2018M_2 |“|“_{{\bf c};w_c} + |“|“ f_1;f_2 |“|“_{{\bf c};w_b} + \|\Lambda_1;\Lambda_2\|_{{\bf c};w_{c+b}})$, proving continuity under perturbations.
• The convergence result holds for a broader class of renormalisation procedures than the original BPHZ, including those with non-trivial derivative corrections.
• The method bypasses the need for intricate algebraic manipulations of iterated integrals and cumulants, relying instead on spectral gap and induction.
• The framework allows for a direct treatment of the first-order Malliavin derivative of the model, embedding it naturally into the structure via the regularity-integrability extension.