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Stochastic heat flow by moments

Li-Cheng Tsai

Abstract

The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), specifically at the critical dimension of two and near the critical temperature. The prior work Caravenna Sun Zygouras (2023) established the first construction of finite-dimensional distributions by demonstrating the universal (model-independent) convergence of discrete polymers. In this work, we present a new, independent approach to the SHF. We formulate the SHF as a continuous process and provide a set of axioms for its characterization. We establish both the uniqueness and existence of this process under our new formulation, with a key feature of these axioms being the matching of the first four moments.

Stochastic heat flow by moments

Abstract

The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), specifically at the critical dimension of two and near the critical temperature. The prior work Caravenna Sun Zygouras (2023) established the first construction of finite-dimensional distributions by demonstrating the universal (model-independent) convergence of discrete polymers. In this work, we present a new, independent approach to the SHF. We formulate the SHF as a continuous process and provide a set of axioms for its characterization. We establish both the uniqueness and existence of this process under our new formulation, with a key feature of these axioms being the matching of the first four moments.

Paper Structure

This paper contains 16 sections, 14 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Moments, Lindeberg's method, proof ideas
  3. Convergence
  4. Polymer measure, conditional GMC, noise
  5. Corollaries, generalizations
  6. Some related literature
  7. Notation and tools
  8. Delta-Bose semigroup
  9. Chapman--Kolmogorov equation and polymer measure
  10. Axiomatic characterization, proof of Theorem \ref{['t.main.']}
  11. Lindeberg's method
  12. Proof of Lemma \ref{['l.key']}
  13. Convergence, proof of Proposition \ref{['p.conv']}
  14. Notation and tools
  15. Checking the limiting properties
  16. ...and 1 more sections

Key Result

Theorem 1.3

If $Z$ and $\widetilde{Z}$ are SHF($\theta$), they are equal in law.

Figures (3)

  • Figure 1: First example
  • Figure 2: Second example
  • Figure 3: Simplified diagram

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Corollary 1.6
  • Proposition 1.7: Theorem 1.6 in gu2021moments
  • Corollary 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 18 more