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Martingale measure associated with the critical $2d$ stochastic heat flow

Makoto Nakashima

TL;DR

The paper proves that the critical $2d$ stochastic heat flow $ig\\mathscr{Z}^ hetaig$ obtained from a discrete stochastic heat equation in the critical window is a continuous semimartingale and satisfies a Walsh-type martingale measure framework. By constructing the measure-valued process $ig\\mathscr{Z}^{ heta, ext{φ}}ig}$ and its martingale part, the authors derive a precise quadratic-variation formula that captures the regularity of the flow. The work combines detailed moment analysis of partition functions via chaos expansions, intricate combinatorial arguments on intersections and stretches, and careful renormalization to obtain tightness and convergence of the finite-dimensional distributions to a unique limit. This framework provides a rigorous pathwise description of fluctuations in the critical two-dimensional directed polymer setting and connects to SPDE methods for measure-valued processes. The results enrich the understanding of the critical regime by linking variance, higher moments, and martingale structure to the underlying random environment through explicit kernels such as $G_ heta$ and convolutions with heat kernels.

Abstract

In [CSZ23], the authors proved the convergence of the finite dimensional time distribution of the rescaled random fields derived from the discrete stochastic heat equation of $2d$-directed polymers in random environment in the critical window. The scaling limit is called critical $2d$ stochastic heat flow (SHF). In this paper, we will show that the critical $2d$ SHF is a continuous semimartingale. Moreover, we will consider the martingale problem associated with the critical $2d$ SHF in a similar fashion to the super Brownian motion which is one of the well-known measure valued process. Also, we define the martingale measure associated with the critical $2d$ SHF in the sense of [Wal86, Chapter 2]. The quadratic variation of the martingale measure gives information of the regularity of the critical $2d$ SHF.

Martingale measure associated with the critical $2d$ stochastic heat flow

TL;DR

The paper proves that the critical stochastic heat flow obtained from a discrete stochastic heat equation in the critical window is a continuous semimartingale and satisfies a Walsh-type martingale measure framework. By constructing the measure-valued process and its martingale part, the authors derive a precise quadratic-variation formula that captures the regularity of the flow. The work combines detailed moment analysis of partition functions via chaos expansions, intricate combinatorial arguments on intersections and stretches, and careful renormalization to obtain tightness and convergence of the finite-dimensional distributions to a unique limit. This framework provides a rigorous pathwise description of fluctuations in the critical two-dimensional directed polymer setting and connects to SPDE methods for measure-valued processes. The results enrich the understanding of the critical regime by linking variance, higher moments, and martingale structure to the underlying random environment through explicit kernels such as and convolutions with heat kernels.

Abstract

In [CSZ23], the authors proved the convergence of the finite dimensional time distribution of the rescaled random fields derived from the discrete stochastic heat equation of -directed polymers in random environment in the critical window. The scaling limit is called critical stochastic heat flow (SHF). In this paper, we will show that the critical SHF is a continuous semimartingale. Moreover, we will consider the martingale problem associated with the critical SHF in a similar fashion to the super Brownian motion which is one of the well-known measure valued process. Also, we define the martingale measure associated with the critical SHF in the sense of [Wal86, Chapter 2]. The quadratic variation of the martingale measure gives information of the regularity of the critical SHF.

Paper Structure

This paper contains 26 sections, 32 theorems, 241 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Setting and known results
  3. Measure valued process
  4. Martingale measure
  5. Organization of the paper
  6. Variance and its limit
  7. Proofs of Theorem \ref{['thm:ContVer']} and Theorem \ref{['thm:SHFMP']}
  8. $\mathsf{Z}_{N;\cdot}^{\phi}$ as measure valued process
  9. Continuity of $\mathscr{Z}^{\vartheta,\phi}$
  10. Proof of \ref{['item:Crel1-1']} for $M^{N,\phi}(\psi)$
  11. Proof of \ref{['item:Crel1-2']} for $M^{N,\phi}(\psi)$
  12. Proof of \ref{['item:CrelM2']} for $M^{N,\phi}_\cdot(\psi)$
  13. Martingale $\mathscr{M}_t^{\vartheta,\phi}(\psi)$
  14. Proof of Lemma \ref{['lem:martingaleZ']}
  15. Proof of Lemma \ref{['lem:QuadConvL2']}
  16. ...and 11 more sections

Key Result

Theorem 1.3

(The local limit theorem Spi76, LL10) Let $q_n$ be the transition probability kernel defined as above. Then, we have for $n\in \mathbb{N}$, $x\in{\mathbb{Z}}^2$, where is the Gaussian density on ${\mathbb{R}}^2$ with mean $0$ and variance $t I$.

Figures (2)

  • Figure 1: An image of the graph associated with $\mathtt{s}$ and $\widetilde{T}_N(\mathtt{s},s,t)$. Curly lines represent wights $U$ and solid lines represent weights $q$.
  • Figure 2:

Theorems & Definitions (78)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 68 more