On the role of positivity preservation for high order approximations of the Dean--Kawasaki equation

Technical Summary: Positivity Preservation in High-Order Approximations of the Dean–Kawasaki Equation

Problem Statement
The Dean–Kawasaki equation describes the evolution of the empirical measure of $N$ independent Brownian particles. While exact at the microscopic level as a martingale solution for an atomic random measure, it is too singular to be interpreted as a density-valued Stochastic Partial Differential Equation (SPDE) due to the non-Lipschitz square-root noise coefficient and scaling supercriticality. To approximate the particle system with a continuum model, regularization is required.

A critical challenge in this approximation is positivity preservation. Since the empirical density must be non-negative, an ideal numerical or analytical approximation should preserve this property. However, existing high-order methods face a trade-off:

1. Finite-difference approaches (e.g., [CF23, CFIR26]) achieve high-order weak convergence in high-density regimes but fail to preserve positivity due to the nonlocal nature of the divergence operator, allowing solutions to take negative values.
2. Positivity-preserving approaches (e.g., [DKP24, DJP26]) truncate the noise or regularize the singularity to ensure non-negativity but suffer from weaker convergence rates or require strong noise truncation that limits approximation quality.

This paper investigates the specific impact of violating positivity preservation on the weak approximation error, particularly in low-density regimes where the initial density may vanish locally. It asks whether non-positivity-preserving approximations can achieve superpolynomial convergence rates and how the error behaves when the initial density is not bounded away from zero.

Methodology
The authors analyze a spectral regularization of the Dean–Kawasaki equation:
$$du_t = \frac{1}{2}\Delta u_t dt + \frac{1}{\sqrt{N}} \nabla \cdot \left( K_\varepsilon * (f(u_t^+) dW_t) \right)$$
where $K_\varepsilon$ is a mollifier with compactly supported Fourier transform, $u_t^+ = \max(u_t, 0)$, and $f$ mimics the square-root noise (e.g., $f(x)=\sqrt{x}$). Unlike previous positivity-preserving schemes, the Wiener process $W$ is not truncated, and the spectral regularization acts on the entire noise coefficient. This choice allows the solution $u_t$ to take negative values, breaking positivity preservation, but offers the advantage of moving the regularization to the test function via convolution, exploiting test function regularity.

The analysis relies on:

1. Duality with the Hamilton–Jacobi–Bellman (HJB) equation: Extending the duality result of [KLvR19] to complex-valued test functions to relate the moment generating function of the SPDE solution to the solution of an infinite-dimensional Kolmogorov backward equation.
2. Laplace Transform Analysis: Measuring the weak error via the difference in Laplace transforms, $|E[e^{\langle u_t, \phi \rangle}] - E[e^{\langle \mu_t, \phi \rangle}]|$, which controls moments of all orders via Cauchy's integral formula.
3. Decomposition of Error: The total error is decomposed into initial approximation error, Fourier cut-off error, and an error term arising specifically from the negative part of the solution ($u_t^-$).
4. Lower Bound Construction: A specific one-dimensional example is constructed where the initial density vanishes on a sub-interval to derive a lower bound on the approximation error, demonstrating the limitations of non-positivity-preserving schemes in low-density regimes.

Key Contributions and Results

1. Uniform-in-Time Weak Error Bounds:
   The paper establishes a uniform-in-time weak error bound measured through the Laplace transform. The error bound consists of three components:

   • Initial approximation error.
   • Fourier cut-off error (dependent on the spectral resolution $\varepsilon$).
   • Negative part error: A term proportional to the probability and magnitude of the solution taking negative values.

2. High-Density Regime ($\rho_{\min} > 0$):
   When the initial density is bounded away from zero ($\rho_0 \ge \rho_{\min} > 0$), the negative part of the solution is exponentially suppressed. Consequently, the spectral approximation achieves a superpolynomial convergence rate for smooth test functions ($\phi \in C^\infty$). Specifically, with an optimal choice of $\varepsilon \sim N^{-1/d}(\log N)^{3/d}$, the error decays as $O(N^{-1-\alpha/d}(\log N)^{3\alpha/d})$ for any $\alpha > 0$. This result holds despite the lack of strict positivity preservation, provided the density is sufficiently high relative to the noise strength.

3. Low-Density Regime ($\rho_{\min} = 0$):
   When the initial density is allowed to vanish, the exponential suppression of the negative part disappears. The upper bound deteriorates to a polynomial rate: $O(N^{-(3\alpha+d)/(2\alpha+d)})$.
   Crucially, the authors provide a lower bound for a one-dimensional example with $\rho_{\min}=0$. They prove that the weak error cannot decay superpolynomially in this regime. The negative part of the solution becomes relatively large, dominating the fluctuations and limiting the approximation quality. This suggests that in low-density regions, approximations that do not preserve positivity are inherently limited in accuracy.

4. Comparison with Existing Methods:

   • In dimensions $d=1, 2$, the upper bound for the non-positivity-preserving scheme in the low-density regime is worse than that of the positivity-preserving SPDE approximation in [DKP24].
   • The results suggest that hybrid schemes (combining high-order spectral approximations in high-density regions with particle-based or positivity-preserving methods in low-density regions) may be the most efficient strategy.

5. Numerical Validation:
   Numerical experiments confirm the theoretical findings. Simulations show that:

   • In high-density regimes, the spectral Galerkin approximation exhibits superpolynomial convergence for smooth observables.
   • In low-density regimes, the probability of the solution becoming negative increases significantly, correlating with the theoretical lower bounds on the error.
   • The loss of positivity is rare when $\rho_{\min}$ is large but frequent when $\rho_{\min} \approx 0$.

Significance and Claims
The paper claims to provide the first analysis of non-positivity-preserving approximations for the Dean–Kawasaki equation that allows for low initial densities. Its primary significance lies in quantifying the cost of violating positivity preservation:

• It demonstrates that positivity preservation is essential for achieving superpolynomial convergence rates when the initial density vanishes.
• It clarifies that while spectral regularization can achieve high-order accuracy in high-density regimes without strict positivity, the "negative excursions" of the solution become a dominant error source in low-density regions, preventing superpolynomial convergence.
• The work distinguishes itself from finite-dimensional approaches (like [CF23]) by utilizing the infinite-dimensional Markovian structure and duality with the HJB equation, allowing for uniform-in-time error bounds and the treatment of complex-valued test functions.

The authors remain modest regarding the universality of these results, noting that the lower bound is derived for a specific one-dimensional example and that the interaction between particles or external potentials (which can create low-density regions dynamically) requires further study, though the current analysis provides a foundational understanding of the approximation limits in such scenarios.