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Moment-based approach for two erratic KPZ scaling limits

Shalin Parekh

TL;DR

This work studies erratic KPZ-type scaling limits for the (1+1)-D multiplicative stochastic heat equation using a moment-based characterization adapted from Tsai. It constructs a four-parameter propagator field and leverages diffusion-in-random-environment representations to identify finite-dimensional limit laws, avoiding heavy machinery from regularity structures. Two main results are obtained: (i) a variance-blowup-type scaling limit for derivative-noise scaling with coefficient $\sigma_p^2=\int_{\mathbb{R}} (\varphi*\varphi^{(2p-1)}(x))^2 dx$, and (ii) a scaling limit for advective noise with coefficient $\gamma_{\mathrm{ext}}^2=\int_{\mathbb{R}} \frac{\varphi*\varphi(a)}{1-\varphi*\varphi(a)} da$, each described through Brownian motion and local-time functionals. The approach yields concise proofs for limit points in very weak topologies, illustrating both the power and limitations of moment-based methods in KPZ-type problems, including the absence of almost-sure convergence in these settings. These results complement prior renormalization and regularity-structure analyses by providing an alternative route to identifying limiting laws via moments and diffusion techniques.

Abstract

A recent paper of Tsai shows how the first few moments of a stochastic flow in the space of measures can completely determine its law. Here we give another proof of this result for the particular case of the one-dimensional multiplicative stochastic heat equation (mSHE), and then we investigate two corollaries. The first one recovers a recent result of Hairer on a ``variance blowup" problem related to the KPZ equation , albeit in a much weaker topology. The second one recovers a KPZ scaling limit result related to random walks in random environments, but in a weaker topology. In these two problems, we furthermore explain why it is hard to directly use the martingale characterization of the mSHE, the chaos expansion, or other known methods. Using the moment-based approach avoids technicalities, leading to a short proof.

Moment-based approach for two erratic KPZ scaling limits

TL;DR

This work studies erratic KPZ-type scaling limits for the (1+1)-D multiplicative stochastic heat equation using a moment-based characterization adapted from Tsai. It constructs a four-parameter propagator field and leverages diffusion-in-random-environment representations to identify finite-dimensional limit laws, avoiding heavy machinery from regularity structures. Two main results are obtained: (i) a variance-blowup-type scaling limit for derivative-noise scaling with coefficient , and (ii) a scaling limit for advective noise with coefficient , each described through Brownian motion and local-time functionals. The approach yields concise proofs for limit points in very weak topologies, illustrating both the power and limitations of moment-based methods in KPZ-type problems, including the absence of almost-sure convergence in these settings. These results complement prior renormalization and regularity-structure analyses by providing an alternative route to identifying limiting laws via moments and diffusion techniques.

Abstract

A recent paper of Tsai shows how the first few moments of a stochastic flow in the space of measures can completely determine its law. Here we give another proof of this result for the particular case of the one-dimensional multiplicative stochastic heat equation (mSHE), and then we investigate two corollaries. The first one recovers a recent result of Hairer on a ``variance blowup" problem related to the KPZ equation , albeit in a much weaker topology. The second one recovers a KPZ scaling limit result related to random walks in random environments, but in a weaker topology. In these two problems, we furthermore explain why it is hard to directly use the martingale characterization of the mSHE, the chaos expansion, or other known methods. Using the moment-based approach avoids technicalities, leading to a short proof.

Paper Structure

This paper contains 6 sections, 15 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['mr']}
  3. Elementary discussion
  4. Proof of the result
  5. Proof of Theorem \ref{['mr2']}
  6. Proof of Theorem \ref{['mr3']}

Key Result

Proposition 1.1

Let $Z_{s,t}$ be a family of $\mathcal{M}(\Bbb R^2)$-valued random variables indexed by $-\infty <s\le t<\infty$ with $s,t\in\Bbb Q,$ all defined on some complete probability space $(\Omega,\mathcal{F},\Bbb P)$. Suppose that the following are true: Then there exists a $C(\Bbb R^4_\uparrow)$-valued random variable $\tilde{Z}$ defined on the same probability space, such that $(Z_{s,t},f) = \int_{\B

Theorems & Definitions (30)

  • Proposition 1.1: (1+1)-dimensional variant of Theorem 1.3 in Ts24
  • Theorem 1.2: Scaling limit of the SHE with derivative noise
  • Theorem 1.3: Scaling limit of the SHE with advective noise
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3: Solving the martingale problem
  • proof
  • Corollary 2.4
  • ...and 20 more