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A Stochastic Reconstruction Theorem on Rectangular Increments with an Application to a Mixed Hyperbolic SPDE

Carlo Bellingeri, Hannes Kern

TL;DR

This work extends the stochastic reconstruction theorem to random germs with rectangular increments adapted to multiparameter filtrations, enabling a robust framework for analyzing mixed stochastic systems in multiple dimensions. The authors develop a wavelet-based reconstruction with a coherent rectangular-increment structure, prove existence and uniqueness of the reconstructed random distributions, and apply the theory to a novel hyperbolic SPDE that combines Walsh integration with Young products. They establish a full fixed-point theory in the Young regime, including composition, multiplication, and integration of germs, yielding local and global well-posedness results. Additionally, the paper clarifies the connection between reconstruction and stochastic sewing in higher dimensions, showing how sewing emerges as a special case and highlighting the broader applicability to products of deterministic and stochastic objects that arise in hyperbolic SPDEs and stochastic analysis.

Abstract

We extend the stochastic reconstruction theorem to a setting where the underlying family of distributions satisfies some natural conditions involving rectangular increments. This allows us to prove the well-posedness of a new class of mixed stochastic partial differential of hyperbolic type which combines standard Walsh stochastic integration and Young products.

A Stochastic Reconstruction Theorem on Rectangular Increments with an Application to a Mixed Hyperbolic SPDE

TL;DR

This work extends the stochastic reconstruction theorem to random germs with rectangular increments adapted to multiparameter filtrations, enabling a robust framework for analyzing mixed stochastic systems in multiple dimensions. The authors develop a wavelet-based reconstruction with a coherent rectangular-increment structure, prove existence and uniqueness of the reconstructed random distributions, and apply the theory to a novel hyperbolic SPDE that combines Walsh integration with Young products. They establish a full fixed-point theory in the Young regime, including composition, multiplication, and integration of germs, yielding local and global well-posedness results. Additionally, the paper clarifies the connection between reconstruction and stochastic sewing in higher dimensions, showing how sewing emerges as a special case and highlighting the broader applicability to products of deterministic and stochastic objects that arise in hyperbolic SPDEs and stochastic analysis.

Abstract

We extend the stochastic reconstruction theorem to a setting where the underlying family of distributions satisfies some natural conditions involving rectangular increments. This allows us to prove the well-posedness of a new class of mixed stochastic partial differential of hyperbolic type which combines standard Walsh stochastic integration and Young products.
Paper Structure (20 sections, 32 theorems, 276 equations)

This paper contains 20 sections, 32 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Stochastic reconstruction and hyperbolic SPDEs
  3. Outline of the paper
  4. Open extensions arising from our work
  5. Preliminaries
  6. Rectangular increments of germs
  7. Wavelet bases
  8. Commuting filtration
  9. The stochastic reconstruction theorem
  10. Hölder spaces for random distributions
  11. Coherent stochastic germs and main statement
  12. Proof of the stochastic reconstruction theorem
  13. Application to a mixed SPDE
  14. Composition and scalar multiplication with functions
  15. Products of processes and distributions
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $F$ be a stochastic germ satisfying the rectangular coherence property: for some $m\ge 2$, $\alpha<0$, $\delta>0$ and $\gamma>-\frac{1}{2}$ such that $\gamma+\delta > 0$ one has the growth condition where ${\boldsymbol\lambda}=(\lambda_1\,, \ldots\,, \lambda_d)\in (0,1]^d$, $\mathbf x= (x_1\,, \ldots ,x_d)$$\mathbf y= (y_1\,, \ldots ,y_d)$ are in compact set, and $\psi_\mathbf y^{{\boldsymbol

Theorems & Definitions (88)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 2.3
  • ...and 78 more