Critical 2D Stochastic Heat Flow (SHF)

• Critical 2D SHF is defined as the universal scaling limit of discrete 2D directed polymers and the renormalized stochastic heat equation, characterized by nontrivial, non-Gaussian behavior.
• It is uniquely determined by axioms including the Chapman–Kolmogorov flow property, independence of increments, and explicit moment formulas linked to the δ–Bose gas.
• Exhibiting maximal noise sensitivity and black noise characteristics, SHF displays extreme intermittency and fractal geometry, with observables decoupling from the underlying white noise.

The critical two-dimensional stochastic heat flow (SHF) is the universal scaling limit of discrete 2D directed polymers and of the mollified multiplicative stochastic heat equation at marginal criticality. It yields a nontrivial, non-Gaussian, measure-valued process solving the singular SPDE ∂ₜu = ½Δu + βuξ for space-time white noise ξ, in dimension two with a one-parameter renormalization of disorder strength. The SHF is uniquely characterized by its Chapman–Kolmogorov flow property, independence, and explicit correlation-moment structure determined by the δ–Bose gas. Most notably, SHF exhibits maximal noise sensitivity and is a "black noise" in the sense of Tsirelson, meaning that its observable random variables asymptotically decouple from any driving noise in the scaling limit.

1. Rigorous Construction and Renormalized Limit

The critical SHF is constructed either via discrete polymer partition functions at intermediate disorder or as the continuum limit of the 2D stochastic heat equation with spatially mollified noise. Precisely, for fixed ε > 0 and mollifier φ, the regularized SHE is

$$\partial_t Z^{\epsilon}_{s,t}(x'; x) = \frac{1}{2} \Delta_x Z^{\epsilon}_{s,t}(x'; x) + \sqrt{\beta_\epsilon}\, Z^{\epsilon}_{s,t}(x'; x) \xi^\epsilon(t, x),$$

with initial condition $Z^{\epsilon}_{s,s}(x'; x) = \delta(x-x')$. Here

• $\xi^\epsilon$ is mollified white noise,
• $$\beta_\epsilon = \frac{2\pi}{|\log \epsilon|} + \frac{\pi}{|\log \epsilon|^2} (\theta - 2\log2 + 2\gamma + 2\iint \Phi(u)\log|u-v|\,\Phi(v)\,du\,dv)$$ (with fixed parameter θ, Euler’s constant γ, and mollifier convolution Φ).

As ε ↓ 0, solutions do not converge to triviality; rather, after suitable logarithmic renormalization, the family $\{Z^{\epsilon}\}_{\epsilon>0}$ admits a unique nontrivial scaling limit, the SHF, with well-defined finite-dimensional laws (Gu et al., 19 Jun 2025).

2. Axiomatic Flow Structure

The SHF is uniquely characterized as an $M_+(\mathbb{R}^4)$-valued process $\{Z_{s,t}\}_{(s,t)\in \mathbb{R}^2_{\le}}$ obeying four main axioms:

• Continuity: $(s,t) \mapsto Z_{s,t}$ is continuous in the vague topology.
• Chapman-Kolmogorov (Convolution): For any $s < t < u$, the mollified product $Z_{s,t} \odot_{u_\ell} Z_{t,u}$ converges to $Z_{s,u}$ in probability as $\ell \to \infty$, irrespective of the mollifier sequence $u_\ell$.
• Independence: $Z_{s,t}, Z_{t,u}$ are independent for disjoint intervals.
• Moment Formulas: For n = 1,2,3,4, moments are given by the semigroup $Q^{(n)}_\theta(r)$ of the attractive δ–Bose gas. In particular,

$\E \left[ \prod_{i=1}^n Z_{s,t}(dx_i, dx'_i) \right] = dx\, dx'\; Q^{(n)}_\theta(t-s; x, x'),$

where for n = 1, $Q^{(1)}_\theta(r;x,x')=p(r,x-x')$ (plain heat kernel), and for higher n, explicit diagrammatic kernels apply (Tsai, 18 Oct 2024).

This axiomatic characterization yields a unique law, independent of the regularization scheme (Gu et al., 19 Jun 2025).

3. Black Noise Property and Noise Sensitivity

The main theorem (Gu et al., 19 Jun 2025) establishes that the SHF is a black noise in the sense of Tsirelson (2004):

• Definition: The filtration generated by SHF increments exhibits vanishing linear (first) Wiener chaos, i.e., $H_1 = \{0\}$, where

$H_1 = \left\{ X \in L^2(\Omega) : \E[X \mid \mathcal{F}_{s,t}] = \E[X \mid \mathcal{F}_{s,u}] + \E[X \mid \mathcal{F}_{u,t}],\,\, \forall s<u<t \right\}.$

• Implication: Any observable depending on finite blocks of the environment becomes asymptotically insensitive to the driving noise in the critical scaling limit; only infinitely high chaos modes contribute to observables (Caravenna et al., 14 Jul 2025).

Technically, this follows via Tsirelson’s variance criterion, built on analytic estimates showing

$\mathrm{Var}(\E[X \mid \mathcal{F}_{s,t}]) \leq C \frac{t-s}{|\log(t-s)|},$

which vanishes superlinearly in small intervals. The underlying kernel decay $W^\theta(r) = O((r|\log r|^2)^{-1})$ enforces this fast decay.

4. Independence from the Underlying White Noise

A direct corollary of the black noise property is the independence of the SHF in the scaling limit from the underlying mollified noise: $$(Z^{\epsilon}, \xi^{\epsilon}) \xrightarrow[\epsilon\to 0]{} (Z, \xi)$$ in joint law, with Z (SHF) and ξ (white noise) being independent random objects (Gu et al., 19 Jun 2025). This is established by tightness, marginal convergence, and the general theory of black-noise independence (cf. Tsirelson 2004, Himwich–Quastel–Zhang 2024 for directed landscape). Any coupling between a white noise and a black noise object must be independent in the scaling limit (Caravenna et al., 14 Jul 2025).

5. Comparison Across Critical/Subcritical/Supercritical SPDE Regimes

The SHF exhibits several features distinguishing criticality from subcritical and supercritical regimes:

• Subcritical (d < 2): With fixed β > 0, the standard 2D SHE admits continuous solutions via Wiener chaos expansions; the associated random field is Gaussian and non-singular (Caravenna et al., 13 Dec 2024).
• Supercritical (d > 2): The multiplicative noise is too singular for classical solution methods; renormalization group techniques or triviality occur.
• Critical (d = 2): The SHF lies strictly outside the Gaussian/Wiener chaos paradigm: the limit object is not a Gaussian multiplicative chaos (no GMC theorem (Caravenna et al., 2022)), moments exhibit double-exponential growth, and all polynomial-energy observables are annihilated in the scaling limit.

The SHF is maximally "noise sensitive"—all finite-chaos observables asymptotically vanish, and resampling any microscopic portion of the disorder produces statistically independent macroscopic limits. This is tightly connected to other black noise objects such as the Brownian web, critical percolation scaling limits, and the directed landscape (Gu et al., 19 Jun 2025).

6. Intermittency, Singularity and Multifractal Structure

The SHF displays extreme intermittency and singularity:

• Spatial singularity: The one-time marginals are almost surely singular with respect to Lebesgue measure; local mass on vanishing balls decays faster than volume, with

$$\lim_{\epsilon\to 0} Z_t(B(x, \epsilon)) / \mathrm{Vol}(B(x, \epsilon)) = 0$$

for almost every x (Liu et al., 18 Oct 2024, Caravenna et al., 8 Apr 2025).

• High-moment growth: The h-th moment on a shrinking ball B(0,ε) scales as

$\E[Z_t(B(0, \epsilon))^h] \sim [\pi \epsilon^2]^h \cdot [\log(1/\epsilon)]^{h(h-1)/2}$

with rapid, super-polynomial intermittency. The observed logarithmic corrections suggest log-multifractality (Liu et al., 18 Oct 2024, Ganguly et al., 29 Jul 2025).

• No GMC: Strict lower bounds on moments disallow any representation as exponential of Gaussian field, excluding SHF from the GMC universality (Caravenna et al., 2022).

This intermittency is underpinned by diagrammatic (collision) expansions, where all Brownian trajectories collide maximally, leading to emergence of extreme peaks at fine scales. The double-exponential growth of high moments is a signature of enhanced intermittency in critical SPDEs (Ganguly et al., 29 Jul 2025).

7. Impact and Connections to Broader SPDE Theory

The critical 2D SHF sets a precedent for the paper of singular SPDEs at criticality:

• Non-Gaussian universality: SHF provides the canonical candidate for the solution to the ill-posed multiplicative 2D SHE, yielding a universal non-Gaussian measure-valued process (Caravenna et al., 11 Nov 2025).
• Black noise paradigm: The intrinsic noise sensitivity alters the statistical independence, leading to independence from underlying noise and links to other black noise systems (Gu et al., 19 Jun 2025, Caravenna et al., 14 Jul 2025).
• Fractal geometry and extinction: The measure concentrates on a rare fractal set, decays to zero locally in the long-time limit, and possesses negative Hölder regularity, sitting just outside of the scope of regularity structures (Caravenna et al., 8 Apr 2025).
• Methodological implications: The analytic tools—chaos expansions, diagrammatic renewal structures, variance decay, and coarse-graining—developed in SHF underpin the rigorous paper of critical singular SPDEs, multifractality, and intermittency, with direct relevance for higher-dimensional KPZ universality (Caravenna et al., 13 Dec 2024, Caravenna et al., 11 Nov 2025).

The SHF remains a focal object for exploring universality, fractal geometry, intermittent behavior, and noise sensitivity in critical SPDEs, with open problems including the full characterization of its multifractal spectrum, the construction of continuum polymer measures, and the rigorous identification of limiting fields for the 2D KPZ equation.