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A semigroup approach to the reconstruction theorem and the multilevel Schauder estimate for singular modelled distributions

Masato Hoshino, Ryoji Takano

TL;DR

The paper develops a semigroup-based approach to singular modelled distributions within Hairer’s regularity-structure framework, proving reconstruction and multilevel Schauder estimates for the singular setting. By defining Besov norms via a G-type semigroup {Q_t} and incorporating temporal weights, it extends the analytic core to regions with boundary/boundary-like singularities and non-translation-invariant operators. The authors provide an alternative, semigroup-driven derivation of reconstruction and MSE with global bounds, and demonstrate a locally well-posed PAM theory in this context, including a spacetime renormalization that yields a convergent renormalized solution for a PAM with spatially varying diffusion. These results enable local-in-time analysis of subcritical singular SPDEs with boundaries or nonuniform coefficients and establish convergence of mollified models to renormalized limits, broadening applicability to non-translation-invariant problems in stochastic PDEs.

Abstract

We extend the semigroup approach used in [23,21] to provide alternative proofs of the reconstruction theorem and the multilevel Schauder estimate for singular modelled distributions. As an application of them, we construct the local-in-time solution of the two dimensional parabolic Anderson model with a non-translation invariant differential operator.

A semigroup approach to the reconstruction theorem and the multilevel Schauder estimate for singular modelled distributions

TL;DR

The paper develops a semigroup-based approach to singular modelled distributions within Hairer’s regularity-structure framework, proving reconstruction and multilevel Schauder estimates for the singular setting. By defining Besov norms via a G-type semigroup {Q_t} and incorporating temporal weights, it extends the analytic core to regions with boundary/boundary-like singularities and non-translation-invariant operators. The authors provide an alternative, semigroup-driven derivation of reconstruction and MSE with global bounds, and demonstrate a locally well-posed PAM theory in this context, including a spacetime renormalization that yields a convergent renormalized solution for a PAM with spatially varying diffusion. These results enable local-in-time analysis of subcritical singular SPDEs with boundaries or nonuniform coefficients and establish convergence of mollified models to renormalized limits, broadening applicability to non-translation-invariant problems in stochastic PDEs.

Abstract

We extend the semigroup approach used in [23,21] to provide alternative proofs of the reconstruction theorem and the multilevel Schauder estimate for singular modelled distributions. As an application of them, we construct the local-in-time solution of the two dimensional parabolic Anderson model with a non-translation invariant differential operator.
Paper Structure (18 sections, 21 theorems, 147 equations)

This paper contains 18 sections, 21 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted Besov space
  4. Temporal weights
  5. Reconstruction of singular modelled distributions
  6. Regularity structures and models
  7. Singular modelled distributions
  8. Reconstruction Theorem
  9. Multilevel Schauder estimate
  10. Regularizing kernels
  11. Compatible models and multilevel Schauder estimate
  12. Parabolic Anderson model
  13. Preliminaries
  14. Regularity structure associated with PAM
  15. Convolution operators
  16. ...and 3 more sections

Key Result

Lemma 2.6

Let $w$ be a $G$-controlled weight. For any $\alpha\ge0$ and $\beta\in[0,\mathfrak{s}_1)$, there exists a constant $C$ such that, for any $t\in(0,1]$ and $x\in\mathbb{R}^d$ we have and

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • proof
  • Proposition 2.8
  • ...and 42 more