Dean Conjecture in Fluctuating Hydrodynamics

• Dean Conjecture is a hypothesis in fluctuating hydrodynamics suggesting that the empirical density of diffusing particles follows a nonlinear Dean–Kawasaki SPDE with hydrodynamic drift and square-root noise.
• Key formulations include various regularizations—such as renormalized kinetic, discretized, and inertial models—that ensure well-posedness while enforcing physical constraints like positivity and mass conservation.
• Rigorous studies show that while the naive white-noise DK equation is ill-posed, properly regularized models can quantitatively approximate particle fluctuations in both independent and interacting systems.

In fluctuating hydrodynamics, the Dean Conjecture is the claim that the empirical density of many diffusing particles is governed by, or well approximated by, a Dean–Kawasaki (DK) stochastic partial differential equation whose drift matches the hydrodynamic limit and whose conservative noise has amplitude proportional to the square root of the local density. In this formulation, the DK equation is intended to provide the correct nonlinear fluctuation description of conservative microscopic dynamics, including interacting systems and regimes near criticality. The modern literature presents a sharply differentiated picture: the unregularized white-noise equation is highly singular and, in key formulations, ill-posed, whereas several regularized or discretized DK models admit rigorous well-posedness, preserve physical constraints such as positivity and mass conservation, and quantitatively approximate particle fluctuations in specific regimes (Djurdjevac et al., 2022, Wang et al., 2022, Cornalba et al., 2023, Schiavo et al., 16 Jan 2025).

1. Historical formulation and conceptual content

The classical DK equation was argued formally by Dean and Kawasaki as an SPDE for the empirical density of finitely many Langevin or Brownian particles. In the physics formulation adopted by recent mathematical work, the conjectural statement is that the mesoscopic density fluctuations of conservative particle systems are described by a conservative stochastic PDE whose deterministic part is the hydrodynamic drift and whose fluctuation term is of square-root type in the density (Cornalba et al., 2018, Wang et al., 2022).

For independent particles on the torus, the conjecture can be stated in its most direct form. Let $d\geq 1$ and let $X^1,\dots,X^N$ be independent standard $d$-dimensional Brownian motions on $\mathbb T^d$, with empirical measure

$\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$

Formally, $\mu^N$ should satisfy the DK SPDE without interactions

$$\partial_t \mu^N=\tfrac12\Delta \mu^N+\tfrac1{\sqrt N}\,\nabla\cdot\big(\sqrt{\mu^N}\,\xi\big), \qquad \mu_0^N=\tfrac1N\sum_{i=1}^N\delta_{x^i},$$

with $\xi$ a vector-valued space-time white noise (Djurdjevac et al., 2022).

For interacting systems, the fluctuating-hydrodynamic version includes a conservative nonlinear drift. A representative Stratonovich form studied for singular mean-field interactions is

$$\mathrm{d}\rho = \Delta\rho\,\mathrm{d}t - \nabla\cdot\big(\rho\,V(t)\ast\rho\big)\,\mathrm{d}t - \nabla\cdot\big(\sqrt{\rho}\circ \mathrm{d}W^F\big),$$

where $V=-\nabla \mathcal U$ and the noise is colored in space and white in time (Wang et al., 2022).

The central conceptual point is therefore not merely the existence of an SPDE with conservative noise, but the claim that such an SPDE is the correct nonlinear fluctuation description of microscopic conservative dynamics. The difficulty is that the original white-noise equation is singular precisely in the terms that encode fluctuation–dissipation.

2. Canonical DK equations and rigorous formulations

Because the product $X^1,\dots,X^N$0 and its divergence are not classically defined, rigorous work uses martingale or kinetic formulations rather than pointwise SPDE calculus. In the independent-particle case, the DK martingale problem requires that for every $X^1,\dots,X^N$1,

$X^1,\dots,X^N$2

is a martingale with quadratic variation

$X^1,\dots,X^N$3

Konarovskyi–von Renesse identify the unique solution in law with the empirical measure of the $X^1,\dots,X^N$4 independent Brownian motions, giving an exact reference dynamics against which approximations can be compared (Djurdjevac et al., 2022).

In the interacting setting with singular Itô correction near $X^1,\dots,X^N$5, recent work replaces classical weak solutions by renormalized kinetic solutions. The kinetic function is

$X^1,\dots,X^N$6

and the associated parabolic defect measure is

$X^1,\dots,X^N$7

This framework intertwines conservative diffusion, nonlocal interaction drift, correlated noise, and the singular Itô correction, while encoding mass conservation and Fisher-information control (Wang et al., 2022).

A further important formulation arises in inertial particle models. Starting from second-order Langevin dynamics on the one-dimensional torus, one introduces a regularized empirical density $X^1,\dots,X^N$8 and momentum density $X^1,\dots,X^N$9 by mollifying Dirac masses with a von Mises kernel. The resulting closed regularized DK model is a two-component SPDE for density and momentum, and is analyzed in mild form on $d$0 with a $d$1-Wiener noise whose covariance is induced by the mollifier (Cornalba et al., 2018).

These formulations already indicate that the conjecture is not about a single literal equation. Rather, the rigorous program concerns which interpretation or regularization of the DK structure preserves the conservative fluctuation mechanism while remaining mathematically meaningful.

3. Ill-posedness, triviality, and the limits of the raw white-noise model

A decisive development is the demonstration that the pure-noise Dean–Kawasaki equation with bounded conservative drift is not a valid continuum model in the measure-valued martingale sense. For

$d$2

the 2025 ill-posedness result proves that when $d$3 and $d$4 is bounded Borel measurable, there are no continuous $d$5-valued martingale solutions for any initial condition; in particular, the pure-noise case $d$6 admits no solutions (Schiavo et al., 16 Jan 2025).

The mechanism is structural. The martingale quadratic variation is exactly

$d$7

and one can choose a nonnegative test function with unit slope on the support of the time-averaged marginal measure, forcing Brownian variance growth proportional to the conserved mass. Bounded drift cannot compensate for the resulting negative fluctuations without contradicting positivity of $d$8 (Schiavo et al., 16 Jan 2025).

This nonexistence result is described as sharp. Solutions are known for certain unbounded singular drifts, and these solutions are atomic, giving reversible coalescing–fragmenting Wasserstein-type dynamics rather than a genuinely regular density evolution (Schiavo et al., 16 Jan 2025). The same paper also recalls the $d$9 regime with smooth bounded drift potentials, where existence occurs only for quantized atomic initial data and the solution coincides with the empirical measure of a finite-particle Langevin system. This is the sense in which earlier work identified an ill-posedness-versus-triviality dichotomy.

The regularization literature had already emphasized the same obstacle. The classical DK equation was described as mathematically open except in very special cases, and the purely diffusive setting was noted to exhibit triviality: for the deterministic drift $\mathbb T^d$0, the only solution is a trivial atomic solution for integer $\mathbb T^d$1, and no solution exists for non-integer $\mathbb T^d$2 (Cornalba et al., 2018). The weakly interacting fluctuation work similarly states that the original continuum DK equation is highly singular, ill-posed, and not renormalizable by regularity structures or paracontrolled distributions (Cornalba et al., 2023).

A common misconception is therefore that the Dean Conjecture has been confirmed for the literal white-noise SPDE. The available results support the opposite conclusion: the naive continuum white-noise equation with bounded drift is not the correct rigorous object, and any viable realization must either regularize the noise or introduce singular drift mechanisms that preserve atomic structure.

4. Regularized realizations of the DK program

The mathematically successful realizations of the conjecture are regularized, discretized, or kinetically reformulated. They differ in regime and technical apparatus, but all preserve the conservative architecture of the DK equation.

Paper	Model class	Main conclusion
(Cornalba et al., 2018)	Regularized inertial DK model for weakly interacting particles	High-probability local well-posedness; small-noise perturbation of an undamped McKean–Vlasov system
(Djurdjevac et al., 2022)	Nonlinear regularized DK SPDE with Itô gradient noise	Strong well-posedness, comparison principle, positivity, mass conservation, entropy estimate, weak-error bound
(Cornalba et al., 2023)	Formally discretized conservative DK equation	Fluctuation law approximates particle fluctuations up to arbitrarily high order in $\mathbb T^d$3 and a discretization error
(Wang et al., 2022)	DK equation with singular nonlocal interactions and correlated noise	Existence of probabilistic weak renormalized kinetic solutions; pathwise uniqueness and strong well-posedness under stronger assumptions

In the Itô regularization for independent particles, the singular square root is replaced by a Lipschitz $\mathbb T^d$4-approximation $\mathbb T^d$5, the noise is truncated in Fourier space at frequencies $\mathbb T^d$6, and the atomic initial data are mollified by a smooth approximation of the identity. The approximate SPDE is

$\mathbb T^d$7

A stochastic parabolicity, or smallness, condition is required: $\mathbb T^d$8 and in the DK specialization this becomes the coercivity condition

$\mathbb T^d$9

Under this condition there is a unique strong solution, together with positivity and mass conservation (Djurdjevac et al., 2022).

The same work proves a comparison principle: if $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$0 almost everywhere, then almost surely $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$1 for all $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$2. Positivity follows when $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$3, and mass conservation is obtained by testing against the constant function $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$4 (Djurdjevac et al., 2022). These properties are emphasized because they preserve the physical interpretation of the solution as a stochastic density.

The inertial regularization starts from second-order Langevin dynamics with pairwise interaction and introduces a mollified density and momentum. The microscopic noise is then replaced by a regular conservative noise whose covariance is inherited from the mollifier, and the model is closed using an interaction approximation, a low-temperature closure for the momentum flux, and a propagation-of-chaos estimate. The resulting SPDE is a small-noise stochastic perturbation of a deterministic moment system, and existence and uniqueness hold locally in time with high probability when $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$5 is sufficiently large under a joint scaling $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$6 (Cornalba et al., 2018).

The discretized weak-interaction theory adopts a different regularization philosophy. Space is discretized on a uniform grid, and the conservative DK noise is implemented as a finite-dimensional SDE system whose discrete quadratic variation matches the particle martingale structure. This avoids continuum renormalization issues by truncating high frequencies at the grid scale while retaining the fluctuation-dissipation structure (Cornalba et al., 2023).

5. Rigorous validations of the conjecture

The strongest quantitative validation presently available in the independent case is the weak-error analysis for Laplace observables

$\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$7

The key device is Laplace duality. If $\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$8 solves

$\mu_t^N=\frac1N\sum_{i=1}^N \delta_{X_t^i}.$9

then its Cole–Hopf transform solves the heat equation and

$\mu^N$0

Applying Itô’s formula to the regularized SPDE solution and controlling the resulting error terms by positivity, Parseval’s identity, mass conservation, and an entropy inequality yields a weak error of order

$\mu^N$1

between the particle system and the nonlinear SPDE approximation (Djurdjevac et al., 2022).

This rate is dimension dependent. The paper states that in $\mu^N$2 it is $\mu^N$3, in $\mu^N$4 approximately $\mu^N$5, and that it deteriorates with $\mu^N$6 thereafter (Djurdjevac et al., 2022). The same analysis proves an entropy inequality controlling

$\mu^N$7

which provides the Fisher-information estimate needed for Fourier truncation errors.

For weakly interacting diffusing particles, the discretized DK model gives a different kind of validation. The law of the density fluctuations predicted by the discretized conservative DK SPDE approximates the law of the fluctuations of the original particle system in suitable weak metrics, with an error that is of arbitrarily high order in the inverse particle number and a discretization error (Cornalba et al., 2023). The analysis is based on a martingale/duality method, a backward evolution for test functions, and a linearization of nonlinear interaction terms through Fourier decomposition. The paper emphasizes that the DK approximation is far more accurate than linearized Gaussian fluctuation models in the weakly interacting regime.

The interacting singular-kernel theory validates a further portion of the conjectural program at the PDE level. Under the Ladyzhenskaya–Prodi–Serrin condition

$\mu^N$8

there exists a probabilistic weak renormalized kinetic solution to the Dean–Kawasaki equation with singular interactions and correlated noise. Under the additional integrability condition

$\mu^N$9

pathwise uniqueness holds and the equation is probabilistically strong (Wang et al., 2022).

That work also establishes well-posedness of the fluctuating Ising–Kac–Kawasaki conservative SPDE

$$\partial_t \mu^N=\tfrac12\Delta \mu^N+\tfrac1{\sqrt N}\,\nabla\cdot\big(\sqrt{\mu^N}\,\xi\big), \qquad \mu_0^N=\tfrac1N\sum_{i=1}^N\delta_{x^i},$$0

placing the phenomenological SPDE proposed for Kawasaki dynamics on a rigorous PDE footing (Wang et al., 2022). The paper explicitly describes this as a step toward the Giacomin–Lebowitz–Presutti conjecture on nonlinear fluctuations of Kawasaki dynamics, while also stating that the full microscopic-to-SPDE convergence remains open.

6. Present status, scope, and open directions

The current state of the Dean Conjecture is neither a blanket confirmation nor a blanket refutation. What has been ruled out is the naive continuum white-noise DK equation with bounded drift as a measure-valued martingale model (Schiavo et al., 16 Jan 2025). What has been established are several mathematically controlled surrogates that retain the conservative square-root-noise structure and capture particle fluctuations in specific regimes.

For independent particles, the regularized nonlinear Itô DK equation is strongly well posed, preserves positivity and mass conservation, satisfies an entropy estimate, and approximates the particle system with a dimension-dependent weak-error bound on Laplace observables (Djurdjevac et al., 2022). For weakly interacting particles, a discretized conservative DK equation reproduces fluctuation laws up to arbitrarily high order in $$\partial_t \mu^N=\tfrac12\Delta \mu^N+\tfrac1{\sqrt N}\,\nabla\cdot\big(\sqrt{\mu^N}\,\xi\big), \qquad \mu_0^N=\tfrac1N\sum_{i=1}^N\delta_{x^i},$$1 and a discretization error (Cornalba et al., 2023). For singular interacting systems, a renormalized kinetic theory yields existence, uniqueness, and strong well-posedness under LPS-type assumptions, and extends to fluctuating Ising–Kac–Kawasaki dynamics (Wang et al., 2022). For inertial mean-field systems, a regularized DK-type model can be derived from particle dynamics using propagation of chaos and Simon’s compactness criterion, with high-probability local well-posedness in a small-noise regime (Cornalba et al., 2018).

Several open problems are stated explicitly across these works. The independent-particle weak-error theory hinges on Laplace duality and does not yet extend to interacting particles (Djurdjevac et al., 2022). The singular-interaction theory provides the PDE backbone for fluctuating hydrodynamics but leaves the microscopic derivation of the SPDE, and the subsequent derivation of the stochastic Cahn–Hilliard limit near criticality, unresolved (Wang et al., 2022). The regularized inertial theory is local in time, one-dimensional, and dependent on mollification and low-temperature closure (Cornalba et al., 2018). The weak-interaction fluctuation theory leaves open extensions to singular or long-range interactions, unbounded domains, and optimized dependence on $$\partial_t \mu^N=\tfrac12\Delta \mu^N+\tfrac1{\sqrt N}\,\nabla\cdot\big(\sqrt{\mu^N}\,\xi\big), \qquad \mu_0^N=\tfrac1N\sum_{i=1}^N\delta_{x^i},$$2 (Cornalba et al., 2023). The ill-posedness theory itself leaves room for alternative formulations with unbounded singular drifts or other regularizing mechanisms (Schiavo et al., 16 Jan 2025).

A plausible implication is that the rigorous content of the Dean Conjecture now lies in a program of equivalence between microscopic particle fluctuations and appropriately regularized conservative SPDEs, rather than in the literal validity of the original white-noise equation. In that program, positivity, mass conservation, entropy dissipation, kinetic stability, and exact martingale covariance are not auxiliary technicalities; they are the criteria by which a DK-type approximation remains faithful to the particle system it is supposed to describe.