[Paper Review] Locality for singular stochastic PDEs

Ismaël Bailleul, Yvain Bruned|arXiv (Cornell University)|Sep 1, 2021

Stochastic processes and financial applications42 references4 citations

TL;DR

This paper extends the theory of regularity structures to singular stochastic PDEs with non-translation invariant differential operators by introducing state-space-dependent preparation maps and renormalization schemes. It establishes a robust framework for renormalized equations in non-translation invariant settings, generalizing the BPHZ renormalization to spacetime-dependent coefficients via pointwise characters, and proves the existence of solutions under such schemes using adapted reconstruction and algebraic techniques.

ABSTRACT

This work deals with singular stochastic PDEs driven by non-translation invariant differential operators. We describe the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses in particular the use of decorated trees with extended decorations.

Motivation & Objective

• To extend the theory of regularity structures to singular stochastic PDEs involving non-translation invariant differential operators.
• To address the failure of standard BPHZ renormalization in non-translation invariant settings due to the need for function-valued renormalization constants.
• To develop a new renormalization scheme using state-space-dependent preparation maps and characters, replacing constant characters with pointwise functions.
• To prove the existence of solutions to the renormalized system in the non-translation invariant setting using adapted reconstruction and algebraic structures.
• To generalize the BPHZ renormalization to spacetime-dependent coefficients via a character-valued function satisfying a pointwise expectation condition.

Proposed method

• Adapts the heat kernel-based analytic framework from Bailleul & Hoshino’s 'Tourist Guide' to non-translation invariant settings, proving necessary regularity estimates (2.1) for non-radial kernels.
• Introduces a strong preparation map $ R $, which fixes the structure of the regularity structure and preserves homogeneous components, ensuring compatibility with the modelled distribution lift.
• Defines a state-space-dependent character $ ho(x, au) $ satisfying $ ho(x, au) = -E[( ilde{Pi}^{R_ ho} au)(x)] $, generalizing the BPHZ character to non-translation invariant settings.
• Constructs a renormalized interpretation operator $ Pi_ ho = Pi vert_{R_ ho} $, where $ R_ ho $ is a state-dependent preparation map, ensuring the existence of an admissible model.
• Uses a co-action $ ar{ ho} vert_{ au} = ( ho ensor One)ar{ ho} $ to define a group structure on state-dependent characters, preserving composition via $ R_ ho R_{ar{ ho}} = R_{ ho ar{ ho}} $.
• Applies the reconstruction theorem verbatim in the non-translation invariant setting, relying on the robustness of the estimate (2.1) and manifold-compatible proofs from Rinaldi & Sclavi [25].

Experimental results

Research questions

• RQ1 How can the theory of regularity structures be extended to handle singular SPDEs with non-constant, non-translation invariant differential operators?
• RQ2 Why does the standard BPHZ renormalization scheme fail in non-translation invariant settings, and what alternative formulation is required?
• RQ3 Can a state-space-dependent renormalization scheme be constructed that generalizes BPHZ while preserving the algebraic and analytic structure of the theory?
• RQ4 What is the correct generalization of the BPHZ character in non-translation invariant settings, and how is it defined via pointwise expectations?
• RQ5 Does the reconstruction theorem remain valid in non-translation invariant settings, and can its proof be adapted to such geometries?

Key findings

• The paper constructs a state-space-dependent preparation map $ R $ that preserves the homogeneous components of the regularity structure and ensures the existence of a well-defined admissible model $ M^R $.
• It proves that the renormalized system takes the form $ ( abla_{x_0} - L^i_{x_1})u_i = F_i(u, abla_{x_1}u) ho + extstyleigsum_{ au eq 0} ho(x, au) rac{F_i( au)(u, abla_{x_1}u)}{S( au)} $, with $ ho(x, au) = -E[(Pi^{R_ ho} au)(x)] $, generalizing BPHZ.
• The renormalization scheme is shown to be consistent with the group structure of characters via $ R_ ho R_{ar{ ho}} = R_{ ho ar{ ho}} $, where $ ho ar{ ho} = ( ho ensor ar{ ho})ar{ ho} $ pointwise.
• The reconstruction theorem remains valid in the non-translation invariant setting, as the key estimate (2.1) is robust under geometric and analytic perturbations.
• The framework is robust enough to be extended to closed manifolds, provided the reconstruction theorem is adapted via manifold-compatible approaches such as Rinaldi & Sclavi [25].
• The method bypasses the need for extended decorations and provides a direct, algebraically consistent renormalization in non-translation invariant settings, with a clear link to the BPHZ scheme in the translation-invariant limit.