2D Stochastic Heat Flow Insights

• 2D Stochastic Heat Flow is a universal measure-valued process emerging as the scaling limit of the critical 2D stochastic heat equation with multiplicative white noise.
• It is constructed via spatial mollification and discrete polymer regularization, revealing exact logarithmic moment scaling and strong intermittency.
• Its non-Gaussian chaos expansion and singular martingale structure distinguish it from classical superprocesses and Gaussian multiplicative chaos frameworks.

The two-dimensional Stochastic Heat Flow (2D SHF) is a universal, measure-valued stochastic process arising as the scaling limit of the two-dimensional stochastic heat equation (SHE) with multiplicative space-time white noise at the critical disorder strength. It constitutes a fundamental solution to a singular SPDE in its marginal dimension, capturing the phase transition between subcritical (Gaussian) and strong disorder (trivial) regimes. The 2D SHF is characterized by its singularity, exact logarithmic moment and correlation asymptotics, rich intermittency, and structural properties that distinguish it sharply from both classical superprocesses and Gaussian multiplicative chaos frameworks.

1. Definition and Construction

The critical 2D stochastic heat equation is formally written as

$$\partial_t u_t(x) = \frac{1}{2} \Delta u_t(x) + u_t(x)\, \dot W(t,x),$$

where $\dot W$ denotes space-time white noise on $\mathbb{R}^2$. Direct interpretation of this equation is ill-posed since the product $u_t(x)\,\dot W(t,x)$ is singular in $d=2$. Instead, one constructs the 2D SHF through:

(a) Regularization and Renormalization:

• Spatial mollification: Replace $\dot{W}(t,x)$ with $$\dot{W}_\varepsilon(t,x) = (\dot{W}(t, \cdot) * j_\varepsilon)(x)$$ for a mollifier $j$ at scale $\varepsilon$.
• Critical coupling: Set $$\beta_\varepsilon = \sqrt{2\pi} (\log \tfrac{1}{\varepsilon})^{-1/2} (1 + o(1))$$, compensating for the ultraviolet divergence.
• Feynman–Kac representation:

$\dot W$0

where $\dot W$1 is planar Brownian motion.

(b) Discrete Directed Polymer Regularization:

• Replace the noise by i.i.d. disorder on $\dot W$2 and tune inverse temperature $\dot W$3.
• Define the point-to-point partition function

$\dot W$4

where $\dot W$5 is a simple random walk.

Both constructions yield a family of random measures $\dot W$6 converging (as $\dot W$7 or $\dot W$8) to the critical 2D SHF $\dot W$9, a measure-valued stochastic process with almost sure singularity with respect to the Lebesgue measure (Liu et al., 2024, Caravenna et al., 2024, Caravenna et al., 11 Nov 2025).

2. Marginal Law, Moments, and Singularity

At each fixed time $\mathbb{R}^2$0, the one-point marginal $\mathbb{R}^2$1 is almost surely singular with respect to Lebesgue measure: $\mathbb{R}^2$2 This singularity mechanism is encapsulated in the scaling of moments of the random mass $\mathbb{R}^2$3: $\mathbb{R}^2$4 with deterministic first moment $\mathbb{R}^2$5. The exponent $\mathbb{R}^2$6 arises from pairwise collisions in chaos expansions. The process exhibits strong intermittency: while the average mass vanishes with Lebesgue rate, moments diverge polynomially in $\mathbb{R}^2$7, indicating the presence of rare, large peaks (Liu et al., 2024, Caravenna et al., 8 Apr 2025, Caravenna et al., 2024).

3. Moment Structure, Chaos Expansion, and Intermittency

The distributional structure of the 2D SHF is governed by a non-Gaussian chaos expansion, which, for integer $\mathbb{R}^2$8, can be written in terms of $\mathbb{R}^2$9 replicas of planar Brownian motion: $u_t(x)\,\dot W(t,x)$0 The key combinatorial ingredient is that the leading divergence in the $u_t(x)\,\dot W(t,x)$1th moment comes from all $u_t(x)\,\dot W(t,x)$2 pairs, each contributing a $u_t(x)\,\dot W(t,x)$3 factor, as only the diagrams with a single pairwise collision per pair dominate in the $u_t(x)\,\dot W(t,x)$4 limit. The proof employs expansion of the exponential in the Feynman-Kac functional and diagrammatics. The full moment expansion enables rigorous intermittency analysis (Liu et al., 2024, Caravenna et al., 2024).

The process is not a Gaussian multiplicative chaos (GMC): strict lower bounds on higher moments show that the moments fail the GMC factorization property (Caravenna et al., 2022).

4. Regularity, Semimartingale Structure, and Martingale Problem

The 2D SHF can be formulated as a solution of a singular martingale problem:

• For any compactly supported $u_t(x)\,\dot W(t,x)$5 and test function $u_t(x)\,\dot W(t,x)$6, the process

$u_t(x)\,\dot W(t,x)$7

admits the semimartingale decomposition

$u_t(x)\,\dot W(t,x)$8

with $u_t(x)\,\dot W(t,x)$9 a continuous martingale.

• The quadratic variation exhibits a logarithmic scaling in the test function scale, reflecting the spatial singularity:

$d=2$0

with $d=2$1 the heat kernel at scale $d=2$2 (Nakashima, 26 Mar 2025).

The field takes values in negative Hölder/Besov spaces strictly rougher than any locally finite positive-index space, exhibits no atoms, and converges to zero locally as $d=2$3 (Caravenna et al., 8 Apr 2025).

5. Law Characterization, Universality, and Black Noise

The 2D SHF is uniquely determined in law by the following axioms (Tsai, 2024, Caravenna et al., 11 Nov 2025):

1. Continuity: $d=2$4 is almost surely continuous in the vague topology.
2. Chapman–Kolmogorov flow property via mollified convolution.
3. Independent increments over disjoint time intervals.
4. First four moments coincide with the semigroup of the attractive two-dimensional $d=2$5-Bose gas.

This process is the universal scaling limit for a large class of two-dimensional directed polymer models in the critical disorder window. Its law is independent of the specific microscopic details, depending only on second-moment data.

The SHF is a "black noise" in the sense of Tsirelson: no nontrivial linear observable admits additive decomposition over increments, and the SHF is asymptotically independent from the driving white noise in the scaling limit (Gu et al., 19 Jun 2025, Caravenna et al., 14 Jul 2025).

6. Phase Regimes, Intermittency, and Connection to Other Models

The parameter $d=2$6 tunes the disorder strength, mapping the subcritical ($d=2$7), critical ($d=2$8), and supercritical ($d=2$9) regimes:

• Subcritical: Solutions exhibit Edwards–Wilkinson fluctuations and the field is Gaussian with bounded covariance (Dunlap et al., 2024).
• Critical: The SHF emerges, non-Gaussianity and logarithmic multifractality are prominent. Moments of shrinking ball masses grow polynomially in $\dot{W}(t,x)$0 with binomial exponent; for large $\dot{W}(t,x)$1, moments grow double-exponentially as $\dot{W}(t,x)$2 (Ganguly et al., 29 Jul 2025).
• Supercritical: The marginal measure vanishes rapidly, with doubly exponential decay in disorder intensity (Berger et al., 4 Aug 2025).

The SHF cannot be represented as an exponential of a Gaussian field, distinguishing it from Gaussian multiplicative chaos. Nonetheless, there are conditional path-space GMC representations and conjectural connections to continuum polymer measures (Clark et al., 2024).

7. Open Problems and Future Directions

Key open problems and research directions include:

• Explicit SPDE (or pathwise) formulations for the SHF beyond moment identities (Caravenna et al., 11 Nov 2025).
• Geometry of extremal peaks/islands and multifractal spectrum.
• Precise upper/lower tail asymptotics for masses of small sets (Nakashima, 24 Jul 2025).
• The Cole–Hopf renormalization and critical 2D KPZ equation.
• Extension of SHF analysis to other critical or near-critical SPDEs (e.g., $\dot{W}(t,x)$3, 2D KPZ) and universality across marginally relevant models (Caravenna et al., 2024).
• Pathwise construction of polymer measures and their conditional GMC structure (Clark et al., 2024).

The SHF thus serves as a central, explicitly computable process capturing the intermittent, multifractal, and singular phenomena at the marginal critical dimension for stochastic heat flows (Liu et al., 2024, Caravenna et al., 2024, Ganguly et al., 29 Jul 2025, Caravenna et al., 11 Nov 2025, Gu et al., 19 Jun 2025, Tsai, 2024).