Critical 2D Stochastic Heat Flow

• Critical two-dimensional stochastic heat flow is a measure-valued limit of the 2D stochastic heat equation, capturing critical behavior with logarithmic renormalization.
• It arises from directed polymers and mollified equations, yielding a universal random flow with a deterministic first moment and singular fixed-time marginals.
• High-moment analysis reveals double-exponential growth and logarithmic intermittency, highlighting significant challenges in critical SPDE theory.

Critical two-dimensional stochastic heat flow is the measure-valued scaling limit that gives a nontrivial meaning to the stochastic heat equation with multiplicative space-time white noise in the critical spatial dimension $d=2$. In the literature it appears as $SHF^\theta_{s,t}(dx,dy)$ or $\mathscr{Z}_t^\vartheta$, depending on whether one emphasizes the two-parameter flow or a fixed-time marginal. It arises from critically tuned $2+1$-dimensional directed polymers and from mollified two-dimensional stochastic heat equations, has deterministic first moment given by the heat kernel, has an explicit logarithmically singular covariance built from a Dickman-subordinator Green function, and lies beyond existing subcritical singular-SPDE theories (Caravenna et al., 2024).

1. Critical equation and logarithmic scaling window

The formal starting point is the multiplicative stochastic heat equation

$$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$

where $\xi$ is space-time white noise. The equation is critical in dimension $2$ in the renormalization-group sense: under diffusive rescaling, the effective coupling is scale-invariant at first order, so the usual subcritical perturbative frameworks do not apply (Caravenna et al., 2024).

A standard continuum regularization mollifies the noise in space, solves

$$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$

and tunes the coupling to criticality through

$$\beta_{\vartheta,\varepsilon} =\sqrt{\frac{2\pi}{-\log\varepsilon} +\frac{\rho+o(1)}{(-\log\varepsilon)^2}}, \qquad \rho=\pi\vartheta+C.$$

Equivalently, in the directed-polymer discretization one uses the critical window

$$\sigma_N^2 =\frac{1}{R_N}\Big(1+\frac{\theta+o(1)}{\log N}\Big), \qquad R_N\sim \frac{\log N}{4\pi}$$

or, in the lecture-notes normalization, $SHF^\theta_{s,t}(dx,dy)$0 and $SHF^\theta_{s,t}(dx,dy)$1 with a second-order $SHF^\theta_{s,t}(dx,dy)$2 correction (Nakashima, 24 Jul 2025, Caravenna et al., 2024).

The term “critical” refers both to the borderline spatial dimension and to the logarithmic renormalization of the coupling. This should not be confused with a classical function-valued solution theory. At criticality the limit is a random measure-valued flow rather than a random function, and the logarithmic corrections are intrinsic rather than removable artifacts (Caravenna et al., 2024).

2. Construction, flow structure, and characterization

Caravenna–Sun–Zygouras constructed the flow first from critical directed polymers, and Tsai identified the same law as the limit of the mollified stochastic heat equation. The rescaled polymer measures

$SHF^\theta_{s,t}(dx,dy)$3

converge in finite-dimensional distributions to a universal process

$SHF^\theta_{s,t}(dx,dy)$4

which is a random flow of locally finite measures on $SHF^\theta_{s,t}(dx,dy)$5 (Caravenna et al., 2024).

Its first moment is deterministic: $SHF^\theta_{s,t}(dx,dy)$6 and its covariance is explicit: $SHF^\theta_{s,t}(dx,dy)$7 The kernel $SHF^\theta_{s,t}(dx,dy)$8 is built from heat kernels and the weighted Green function $SHF^\theta_{s,t}(dx,dy)$9 of the Dickman subordinator, whose small-time asymptotics are logarithmically singular (Caravenna et al., 2024).

The flow satisfies a Chapman–Kolmogorov property in a regularized sense and has independent increments over disjoint time intervals. Tsai’s axiomatic characterization states that continuity in $\mathscr{Z}_t^\vartheta$0, an appropriate flow property, independence on disjoint intervals, and agreement of mixed moments up to order $\mathscr{Z}_t^\vartheta$1 determine the law of the critical flow (Caravenna et al., 2024).

A complementary construction develops continuum directed polymer measures $\mathscr{Z}_t^\vartheta$2 on path space from the flow. The key step is a Chapman–Kolmogorov relation for $\mathscr{Z}_t^\vartheta$3 using Gaussian smoothing, together with a conditional expectation formula. The resulting path-space measures have first moment equal to Wiener measure and second moments consistent with a conditional Gaussian multiplicative chaos-type interrelationship across the disorder parameter $\mathscr{Z}_t^\vartheta$4 (Clark et al., 2024).

3. Fixed-time marginals: singularity, regularity, and long-time behavior

At fixed time $\mathscr{Z}_t^\vartheta$5, the critical flow defines a random measure $\mathscr{Z}_t^\vartheta$6 on $\mathscr{Z}_t^\vartheta$7. A central spatial result is that this measure is almost surely singular with respect to Lebesgue measure. More precisely,

$\mathscr{Z}_t^\vartheta$8

almost surely, and equivalently the absolutely continuous part vanishes almost surely (Caravenna et al., 8 Apr 2025).

The same work proves that the fixed-time marginal lies almost surely in

$\mathscr{Z}_t^\vartheta$9

hence in every negative Hölder space $2+1$0. This implies absence of atoms. The resulting picture is therefore specific: the measure is singular but non-atomic, and “almost a function” only in the weak distributional sense encoded by negative Hölder regularity (Caravenna et al., 8 Apr 2025).

The proof of singularity proceeds by probing a quasi-critical regime and establishing asymptotic log-normality for the mass of shrinking balls under weak disorder. This suggests that the local structure is governed by rare large fluctuations rather than by a density field. The same analysis yields local convergence to zero in the long-time limit: for every bounded $2+1$1,

$2+1$2

This local vanishing is compatible with the scaling covariance of the flow and with its singular fixed-time geometry (Caravenna et al., 8 Apr 2025).

A common misconception is to treat the critical flow as a rough random density. The fixed-time object is instead a singular random measure. Another is to identify it with Gaussian multiplicative chaos. The literature explicitly states that the SHF is not a Gaussian multiplicative chaos, even though logarithmic correlations and multiplicative-chaos analogies appear in several moment formulas (Caravenna et al., 8 Apr 2025).

4. Mass of balls, lower tails, positivity, and shrinking-scale observables

Because the fixed-time marginal is singular, local analysis is naturally formulated through masses of sets rather than pointwise values. A fundamental observable is

$2+1$3

the total mass transported from a source ball to a target ball. In the polymer interpretation, it is the total critical polymer mass of paths starting in $2+1$4 and ending in $2+1$5 at time $2+1$6 (Nakashima, 24 Jul 2025).

For this observable, the lower tail is sharply constrained. Fix $2+1$7. Then for each $2+1$8, $2+1$9, and $$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$0, there exist constants such that for all integers $$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$1,

$$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$2

As consequences, $$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$3 for every $$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$4, and the ball mass is strictly positive almost surely. More generally, if $$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$5 is nontrivial, then

$$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$6

This resolves an open question on strict local positivity, with Clark–Tsai obtaining an independent proof by different methods (Nakashima, 24 Jul 2025).

At shrinking spatial scales, the moments of normalized ball masses exhibit logarithmic intermittency. For $$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$7,

$$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$8

grows like

$$\partial_t u(t,x)=\tfrac12\Delta u(t,x)+\beta\,\xi(t,x)\,u(t,x), \qquad t>0,\ x\in\mathbb{R}^2,$$9

up to lower-order corrections as $\xi$0. Equivalently,

$\xi$1

This identifies a logarithmic multifractal-type signature in local masses and quantifies the intermittent small-scale geometry already suggested by singularity (Liu et al., 2024).

5. Moment hierarchies, delta-Bose gas structure, and high-moment asymptotics

Single-time moments of the critical two-dimensional stochastic heat equation at criticality were analyzed through the $\xi$2-particle delta Bose gas. For the mollified equation with critical tuning,

$\xi$3

the $\xi$4-point functions converge to a semigroup $\xi$5, where $\xi$6 is the free heat semigroup and $\xi$7 is an explicit interaction operator written as a convergent series over collision diagrams. The corresponding renormalized Hamiltonian is the two-dimensional $\xi$8-body delta Bose gas, and the limiting fixed-time field is non-Gaussian (Gu et al., 2019).

For the critical SHF itself, high moments are far larger than the Gaussian-correlation lower bound $\xi$9. The sharp lower bound proved later states that for sufficiently large $2$0,

$2$1

where $2$2 and $2$3 is smooth, compactly supported, and satisfies $2$4. This matches the late-1990s prediction that the true growth should be double exponential in a linear function of $2$5, and it exponentially improves the previous lower bound. The same work obtains upper-tail bounds

$2$6

for large $2$7 (Ganguly et al., 29 Jul 2025).

The conceptual advance behind these estimates is a new connection between SHF Feynman diagrams and Gaussian free fields on associated graphs. Spatial integrals in the diagrammatic moment formulas become GFF partition functions; by the Matrix–Tree theorem they reduce to weighted spanning-tree sums. This opens an algebraic route to moment estimates and yields, as a byproduct, a monotonicity property of the correlation kernel $2$8 under spatial scaling, derived from the domain Markov property of the GFF (Ganguly et al., 29 Jul 2025).

6. Noise structure, universality class, and open directions

The critical flow is not merely non-Gaussian; it defines a black noise in Tsirelson’s sense. If $2$9 denotes the linear part of the noise in the sense of additive conditional expectations over time intervals, then

$$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$0

As a corollary, the jointly scaled pair $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$1 converges to $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$2 with $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$3 independent of $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$4, even though each prelimit $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$5 is a measurable functional of the mollified white noise $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$6. Thus the critical continuum limit loses any linear footprint of the original driving noise (Gu et al., 19 Jun 2025).

This property clarifies a structural point that is easy to miss: the critical flow should not be viewed as a classical SPDE solution functional of white noise. The lecture notes emphasize that existing singular-SPDE theories handle subcritical equations, whereas the critical $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$7 SHE requires an entirely different construction based on scaling limits, coarse graining, renewal theory, and Lindeberg principles (Caravenna et al., 2024).

Several directions remain open. The proposed logarithmic renormalization of $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$8 toward a $$\partial_t u^{\vartheta,\varepsilon} =\tfrac12\Delta u^{\vartheta,\varepsilon} -\beta_{\vartheta,\varepsilon}u^{\vartheta,\varepsilon}\dot{\mathcal W}^\varepsilon,$$9 KPZ field is still conjectural; identifying the correct diverging normalization $$\beta_{\vartheta,\varepsilon} =\sqrt{\frac{2\pi}{-\log\varepsilon} +\frac{\rho+o(1)}{(-\log\varepsilon)^2}}, \qquad \rho=\pi\vartheta+C.$$0 and proving convergence are unresolved. The exact high-moment asymptotics of $$\beta_{\vartheta,\varepsilon} =\sqrt{\frac{2\pi}{-\log\varepsilon} +\frac{\rho+o(1)}{(-\log\varepsilon)^2}}, \qquad \rho=\pi\vartheta+C.$$1 as $$\beta_{\vartheta,\varepsilon} =\sqrt{\frac{2\pi}{-\log\varepsilon} +\frac{\rho+o(1)}{(-\log\varepsilon)^2}}, \qquad \rho=\pi\vartheta+C.$$2 remain unknown; current results establish the double-exponential scale but not the final exponent. Finer local geometry—multifractality, support dimension, and the precise relation to Liouville-type constructions—also remains open (Nakashima, 24 Jul 2025, Ganguly et al., 29 Jul 2025).

In the current state of the subject, critical two-dimensional stochastic heat flow is best understood as a universal, measure-valued, non-Gaussian random heat flow in the critical dimension: singular at fixed times, strictly locally positive on balls, logarithmically intermittent at small scales, and structurally independent of any residual first-order white-noise component (Caravenna et al., 2024).