2D Stochastic Heat Flow

• 2D stochastic heat flow is a family of random measure-valued processes arising as the critical scaling limit of directed polymers and renormalized stochastic heat equations.
• Its construction uses careful renormalization and mollification to cancel divergences and ensure convergence to a nontrivial random measure.
• The process exhibits intermittency, double-exponential moment growth, and black noise properties, providing insights into critical disorder and free energy asymptotics.

The two-dimensional stochastic heat flow (2D SHF) refers to the family of random measure-valued processes that arise as critical scaling limits of partition functions of directed polymers in a 2D random environment, or as solutions to the stochastic heat equation (SHE) with multiplicative space-time white noise after suitable renormalization. The construction, regularity, and limiting behaviors of the 2D SHF are central to the probabilistic analysis of singular SPDEs in critical dimensions, and they connect deeply to topics such as universality classes, intermittency, Gaussian multiplicative chaos (GMC), integrable systems, and the theory of black noise.

1. Construction and Critical Scaling Limits

The 2D SHF is constructed as the nontrivial scaling limit of either

• (a) the partition function of a two-dimensional directed polymer in random environment (under intermediate disorder scaling), or
• (b) the solution to the two-dimensional stochastic heat equation with multiplicative white noise, after mollification and logarithmic renormalization of the coupling constant.

Given the formal SHE

$$\partial_t u(t,x) = \frac{1}{2}\Delta u(t,x) + \beta\,\xi(t,x)u(t,x), \qquad x \in \mathbb{R}^2,\ t>0$$

where $\xi$ denotes space-time white noise, direct interpretation is obstructed in $d=2$ due to the singular product $\xi u$. Mollification (spatial convolution) leads to

$$\xi_\varepsilon(t,x) = \int_{\mathbb{R}^2} \phi_\varepsilon(x-y) \xi(t,y)dy$$

with an appropriate choice of mollifier $$\phi_\varepsilon(x) = \varepsilon^{-2}\phi(x/\varepsilon)$$.

To obtain a nontrivial limit as $\varepsilon \to 0$, the noise strength is tuned to criticality: $$\beta_\varepsilon = \frac{2\pi}{|\log \varepsilon|} + \frac{2\pi\,\text{fine}}{|\log\varepsilon|^2}$$ with "fine" an adjustment term. This matches and cancels the primary divergent contribution from the Green's function $p(t,0) \sim 1/(2\pi t)$ (Gu et al., 2019, Caravenna et al., 2021).

Alternately, in the discrete directed polymer setting, the critical inverse temperature scaling is

$$\beta_N = \frac{\hat\beta}{\sqrt{R_N}},\qquad R_N \sim \frac{\log N}{4\pi}$$

where $N$ is the discrete system's size, and the scaling window $\hat\beta\to1$ separates subcritical (Gaussian/EW), critical (SHF), and supercritical (strong disorder) regimes (Caravenna et al., 13 Dec 2024, Caravenna et al., 2021, Berger et al., 4 Aug 2025).

The resulting SHF is a process $\mathscr{Z}_{s,t}^\vartheta(dx,dy)$ of random measures on $\mathbb{R}^2$, indexed by disorder strength parameter $\vartheta$, with stationary increments, a Markov property in the time parameter, and meaningful spatial and temporal scaling limits (Tsai, 18 Oct 2024, Nakashima, 26 Mar 2025).

2. Structural and Axiomatic Characterization

The SHF is specified axiomaticaly by the following properties (Tsai, 18 Oct 2024):

• Measure-valued process: For every $s < t$, $\mathscr{Z}_{s,t}$ is a random locally finite measure on $\mathbb{R}^2\times \mathbb{R}^2$.
• Chapman–Kolmogorov/Composition: For $s < t < u$,

$$\mathscr{Z}_{s,u} = \mathscr{Z}_{s,t} \,\circledast\, \mathscr{Z}_{t,u}$$

where $\circledast$ is a suitably defined convolution product integrating the heat kernel semigroup structure at the measure level.

• Independence: Increments over disjoint time intervals are independent.
• Moment Axiom: For test functions $g_i, g'_i \in L^2(\mathbb{R}^2)$,

$$\mathbb{E}\Big[\prod_{i=1}^n \mathscr{Z}_{s,t}(g_i \otimes g'_i) \Big] = \left\langle \bigotimes_{i=1}^n g_i,\, Q^{[n,\vartheta]}(t-s)\bigotimes_{i=1}^n g'_i \right\rangle$$

where $Q^{[n,\vartheta]}(t)$ is the semigroup of the $n$-particle delta Bose gas (Gu et al., 2019, Tsai, 18 Oct 2024). Uniqueness in law holds under these axioms when the moment axiom is enforced for $n=1,2,3,4$.

3. Regularity, Singularity, and Intermittency

The spatial marginals of the SHF are almost surely singular with respect to Lebesgue measure: for fixed $t>0$, the random measure assigns vanishingly small mass to shrinking balls $B(x, \epsilon)$, even after normalizing by their Euclidean volume,

$$\frac{\mathscr{Z}_t(B(x, \epsilon))}{\pi\epsilon^2} \xrightarrow{\;d\;} \exp(\mathcal{N}(0,\sigma^2) - \tfrac{1}{2}\sigma^2)$$

with $\sigma^2 = \log(1+\lambda)$ in a “quasi-critical” regime, demonstrating asymptotic log-normal fluctuations (Caravenna et al., 8 Apr 2025). The density with respect to Lebesgue measure thus vanishes almost everywhere.

Nonetheless, the process is regular in the sense that

$$\mathscr{Z}_t \in \bigcap_{\epsilon>0} \mathcal{C}^{-\epsilon}$$

almost surely (i.e., it is a Schwartz distribution in all negative Hölder spaces but never a function) (Caravenna et al., 8 Apr 2025). There are no atoms, and the measure locally converges to zero in the long time limit, linking to local extinction phenomena in critical polymer models (Caravenna et al., 8 Apr 2025).

Intermittency manifests as explosive high moments for shrinking balls: $$\mathbb{E}[\mathscr{Z}_t(B(x, \epsilon))^h] \sim (\log(1/\epsilon))^{\binom{h}{2}}$$ (Liu et al., 18 Oct 2024). This reflects rare but significant peaks, and indicates multifractal scaling.

4. Moment Structure, Tail Behavior, and Non-Gaussianity

The moment structure of the 2D SHF is sharply non-Gaussian. The moments $m$ of the mass $\mathscr{Z}_t(p)$ assigned to a test function $p$ satisfy, for large $m$,

$$\exp(C m^2) \leq \mathbb{E}[\mathscr{Z}_t(p)^m] \leq \exp(\exp(C' m^2))$$

but the true growth is double-exponential, i.e.,

$$\mathbb{E}[\mathscr{Z}_t(p)^h] \gtrsim \exp(\exp(c_0 h))$$

for some $c_0 > 0$ and all large $h$ (Ganguly et al., 29 Jul 2025). This confirms predictions from the physics literature and distinguishes the SHF from genuine Gaussian Multiplicative Chaos (GMC) measures, which at best display lognormal, not double-exponential, behavior (Caravenna et al., 2022).

The sharp tail asymptotics inherit this double-exponential growth, giving: $$P(\mathscr{Z}_t(p) > z) \approx \begin{cases} \exp(-(\log z)^{(\log\log z)^{1+o(1)}})\quad &\text{upper tail}\ \exp(-c \log z\, \sqrt{\log\log z}) &\text{lower tail} \end{cases}$$ (Ganguly et al., 29 Jul 2025).

5. Markov, Martingale, and Black Noise Properties

The SHF flow is Markovian and admits a continuous semimartingale decomposition when integrated against smooth test functions: for test functions $\varphi, \psi$,

$$Z^{\vartheta, \varphi}_t(\psi) = Z^{\vartheta, \varphi}_0(\psi) + \int_0^t Z^{\vartheta, \varphi}_s\left(\frac{1}{2}\Delta\psi\right) ds + \mathcal{M}^{\vartheta, \varphi}_t(\psi)$$

where $\mathcal{M}^{\vartheta, \varphi}_t(\psi)$ is a continuous local martingale with quadratic variation

$$\langle \mathcal{M}^{\vartheta, \varphi}(\psi) \rangle_t = -\lim_{\delta \to 0} \frac{4\pi}{\log(1/\delta)} \int_0^t \int_{\mathbb{R}^2} [Z^{\vartheta, \varphi}_s(p_\delta(\cdot-z))]^2\psi(z)^2 dz ds$$

where $p_\delta$ is the heat kernel (Nakashima, 26 Mar 2025). The process admits an extension to a Walsh–type martingale measure.

A fundamental finding is that the SHF constitutes a black noise in the sense of Tsirelson: the space of linear observables $H_1$ in $L^2(\Omega)$ is trivial ($H_1 = \{0\}$), meaning the noise cannot be reconstructed by linear means from sub-interval observables (Gu et al., 19 Jun 2025). The scaling limit of the mollified SHE is asymptotically independent of the driving white noise; no linear information persists, highlighting a maximal noise sensitivity (Caravenna et al., 14 Jul 2025, Gu et al., 19 Jun 2025).

6. Conditional Gaussian Multiplicative Chaos and Coupling

While the SHF cannot be globally represented as a Gaussian Multiplicative Chaos with respect to Lebesgue or Wiener measure, the family of path-space polymer measures $M^{\vartheta}_{[s,t]}$ indexed by $\vartheta$ admits a conditional GMC structure: for any $a>0$, forming a multiplicative chaos with respect to $M^{\vartheta}_{[s,t]}$ using an independent Gaussian field of noise strength $a$ recovers the law of $M^{\vartheta+a}_{[s,t]}$,

$$G\left[ M^{\vartheta}_{[s,t]},\, Y^{\vartheta},\, (\xi_i) \right] \stackrel{\text{law}}{=} M^{\vartheta+a}_{[s,t]}$$

where $Y^{\vartheta}$ is an appropriate operator and $(\xi_i)$ are independent centered Gaussians (Clark et al., 21 Jul 2025, Clark et al., 3 Sep 2024).

Two crucial consequences follow:

• For any nonnegative, nontrivial test function, $\mathscr{Z}^{\vartheta}_{s,t}(g \otimes g') > 0$ almost surely.
• In the strong disorder limit ($\vartheta \to \infty$), $\mathscr{Z}^{\vartheta}_{s,t}(g \otimes g') \to 0$ almost surely.

The conditional GMC structure provides a canonical coupling between SHFs at different disorder strengths, echoing related hierarchical models (Clark et al., 3 Sep 2024).

7. Strong Disorder, Extinction, and Free Energy Bounds

As the disorder parameter $\vartheta \to \infty$ (strong disorder regime), the SHF vanishes locally at an explicit doubly-exponential rate: $$\sup_{\varphi \in M_1(B(\sqrt{e^{c_0 t e^{\vartheta}}t}))} \mathbb{E}[\mathscr{Z}_t^{\vartheta}(\varphi) \wedge 1 ] \asymp e^{-c t e^{\vartheta}}$$ for universal $c, c_0 > 0$, confirming optimal collapse as disorder increases (Berger et al., 4 Aug 2025). Analogous results hold for 2D directed polymers in the quasi-critical scaling regime.

For the directed polymer free energy $$\tau_f(\beta) = \lim_{N \to \infty} (1/N) \log Z_N^\beta$$, sharp bounds hold: $$- (c'/\sigma^4) \exp(-\pi/\sigma^2) \leq \tau_f(\beta) \leq -c \exp(-\pi/\sigma^2)$$ with $\sigma^2 = \text{Var}(Z_N^\beta)$ and $c, c'>0$ (Berger et al., 4 Aug 2025).

The proofs exploit coarse-graining, truncated and fractional moment estimates, size-bias change of measure, and a chain of implications connecting local vanishing to global free energy asymptotics.

8. Connections, Open Problems, and Broader Impact

The 2D SHF provides a canonical probabilistic solution concept in the critical dimension for the stochastic heat equation with multiplicative noise, lying beyond the universality of subcritical theories (e.g., Hairer's regularity structures). It is fundamentally distinct from Gaussian multiplicative chaos fields—the higher moments and tail behavior are qualitatively different. Universality questions persist: the SHF is the scaling limit not only for discrete polymers and the mollified SHE, but is robust to the fine structure of the microscopic disorder, as highlighted by its noise insensitivity and black noise properties (Gu et al., 19 Jun 2025, Caravenna et al., 14 Jul 2025, Caravenna et al., 13 Dec 2024).

Key directions for ongoing research include:

• Determining the multifractal spectrum and Hausdorff dimension of the SHF's support.
• Clarifying the geometric structure of peaks and the statistics of extremes.
• Extending the SHF concept to describe continuum KPZ in two dimensions (where the logarithm of the SHF is needed but nontrivial).
• Further analysis of the free energy, both sharp asymptotics and universality among various 2D models.
• Noise theory and couplings—examining the SHF as a black noise, and its implications for Tsirelson's classification of noises in probability.

Table: Core Properties of the 2D SHF

Feature	Property/Behavior	Reference
Construction	Scaling limit of polymer/SHE with critical renormalization	(Caravenna et al., 13 Dec 2024, Berger et al., 4 Aug 2025)
Measure singularity	a.s. singular w.r.t. Lebesgue measure	(Caravenna et al., 8 Apr 2025, Liu et al., 18 Oct 2024)
Moment growth	Double-exponential in order: $\exp(\exp(\Theta(h)))$	(Ganguly et al., 29 Jul 2025, Caravenna et al., 2022)
Black noise	Trivial linear structure; asymptotic independence from driving noise	(Gu et al., 19 Jun 2025, Caravenna et al., 14 Jul 2025)
Conditional GMC	SHFs at different $\vartheta$ coupled by GMC transformation	(Clark et al., 21 Jul 2025, Clark et al., 3 Sep 2024)
Strong disorder	Local vanishing at rate $\asymp e^{-c t e^{\vartheta}}$	(Berger et al., 4 Aug 2025, Clark et al., 21 Jul 2025)
Semimartingale property	Integrated SHF is a continuous semimartingale	(Nakashima, 26 Mar 2025)

The 2D stochastic heat flow thus stands as a central object at the intersection of singular SPDEs, disordered systems, KPZ universality, and contemporary probability theory, characterized by sharp intermittency, multifractality, a highly non-Gaussian structure, and profound noise sensitivity.