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Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality

Alexander Dunlap, Cole Graham

TL;DR

The paper analyzes the two-dimensional stochastic heat equation with scale-dependent, weakly attenuated noise and proves that the system is stable under a renormalization flow that redefines the nonlinearity via a root decoupling function. By connecting pointwise statistics to a forward–backward SDE (FBSDE) and introducing a decoupling flow governed by a parabolic PDE, the authors extend the known results to a broad, $L^{2}$-subcritical class of nonlinearities and show universality of limiting statistics: coarse-grained fields converge in Wasserstein distance to law-identified limits independent of the noise’s fine structure. A key methodological achievement is an induction-on-scales framework that propagates renormalized dynamics through exponential time-scales, yielding both one-point and multipoint asymptotics with quantitative convergence rates. The results illuminate how nonlinear stochastic PDEs in critical dimension exhibit robust, universal behavior under renormalization, with potential implications for related KPZ-type and directed-polymer models in two dimensions.

Abstract

We consider a two-dimensional stochastic heat equation with noise correlated at scale $ρ\ll 1$ and of strength $|\logρ|^{-1/2}σ(v)$ depending nonlinearly on the solution $v$. Under certain conditions, the first author and Gu have shown that the one-point statistics of $v$ converge in law as $ρ\to 0$ to the terminal value of an associated forward-backward SDE. Here, we show that the 2D stochastic heat equation is stable under renormalization with a new effective nonlinearity tied to the decoupling function of the forward-backward SDE. This allows us to extend the pointwise results to a much broader class of nonlinearities. We also show that these limiting pointwise statistics are insensitive to the fine details of the noise, and thus universal.

Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality

TL;DR

The paper analyzes the two-dimensional stochastic heat equation with scale-dependent, weakly attenuated noise and proves that the system is stable under a renormalization flow that redefines the nonlinearity via a root decoupling function. By connecting pointwise statistics to a forward–backward SDE (FBSDE) and introducing a decoupling flow governed by a parabolic PDE, the authors extend the known results to a broad, -subcritical class of nonlinearities and show universality of limiting statistics: coarse-grained fields converge in Wasserstein distance to law-identified limits independent of the noise’s fine structure. A key methodological achievement is an induction-on-scales framework that propagates renormalized dynamics through exponential time-scales, yielding both one-point and multipoint asymptotics with quantitative convergence rates. The results illuminate how nonlinear stochastic PDEs in critical dimension exhibit robust, universal behavior under renormalization, with potential implications for related KPZ-type and directed-polymer models in two dimensions.

Abstract

We consider a two-dimensional stochastic heat equation with noise correlated at scale and of strength depending nonlinearly on the solution . Under certain conditions, the first author and Gu have shown that the one-point statistics of converge in law as to the terminal value of an associated forward-backward SDE. Here, we show that the 2D stochastic heat equation is stable under renormalization with a new effective nonlinearity tied to the decoupling function of the forward-backward SDE. This allows us to extend the pointwise results to a much broader class of nonlinearities. We also show that these limiting pointwise statistics are insensitive to the fine details of the noise, and thus universal.
Paper Structure (56 sections, 72 theorems, 536 equations, 2 figures)

This paper contains 56 sections, 72 theorems, 536 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Main results
  4. The decoupling flow
  5. Examples
  6. Ideas of the proof
  7. The martingale.
  8. Relating the quadratic variation to the martingale.
  9. Extension to longer scales.
  10. Organization
  11. Notational conventions
  12. Acknowledgments
  13. Well-posedness of the FBSDE
  14. SDE solution theory
  15. Local solution theory for the FBSDE
  16. ...and 41 more sections

Key Result

Theorem 1.3

Let $\sigma$ be $L^2$-subcritical. Fix $Q \in (0, 1)$ and define $\widetilde{\sigma} \coloneqq (1 - Q)^{1/2} J_\sigma(Q, \cdot)$ and $\widetilde{\rho} \coloneqq \rho^{1 - Q}$. Then there is a new white noise $\mathrm{d}\widetilde{W}$ such that $\mathcal{G}_{\widetilde{\rho}} v$ is an approximate mil

Figures (2)

  • Figure 1.1: The tree structure of the Brownian motion correlations in thm:mainthm-multipoint: $B^{(i)}$ and $B^{(j)}$ have identical increments until time $p^{(i,j)}$, after which they are independent.
  • Figure 7.1: Schematic illustration of the sets $\mathtt{S}^{(i)}$. The points $z^{(i)} = (T^{(i)}_1,x^{(i)})$ are drawn with colored dots, and the corresponding parabolic cones $\mathtt{E}^{(i)}$ are the regions under the parabolas. In this example, we have $P^{\mathrm{time}}(1) = \{2\}$, $P^{\mathrm{time}}(2) = \{1\}$, and $P^{\mathrm{time}}(3)=\emptyset$, so $P^{\mathrm{time}}_-(1) =\{2\}$ and $P^{\mathrm{time}}_-(2) = P^{\mathrm{time}}_-(3)=\emptyset$. Therefore, to form the set $\mathtt{S}^{(1)}$ (shaded in blue), we remove the time interval $[T_*^{(1,2)},T_1^{2}]$ from $\mathtt{E}^{(1)}$. The sets $\mathtt{S}^{(2)}$ and $\mathtt{S}^{(3)}$ (shaded in red and yellow, respectively) are equal to all of $\mathtt{E}^{(2)}$ and $\mathtt{E}^{(3)}$, respectively.

Theorems & Definitions (150)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Informal
  • Theorem 1.4
  • Theorem 1.5: Informal
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 2.1
  • ...and 140 more