Gaussian fluctuations for the nonlinear stochastic heat equation with drift

Abstract: In this article, we prove the Quantitative Central Limit Theorem (QCLT) for the spatial average of the solution of the nonlinear stochastic heat equation with constant initial condition, driven by space-time Gaussian white noise in dimension 1. The novelty is that the equation contains a drift term. We assume that the drift and diffusion coefficients are twice differentiable with bounded first and second order derivatives. For the proof, we use Malliavin calculus, and the second-order Poincaré inequality due to Vidotto (2020). To estimate the moment of the second Malliavin derivative of the solution, we develop a novel estimate for the product of two heat kernels, which is of independent interest. Finally, we provide the functional result corresponding to this CLT.

• The paper establishes a quantitative central limit theorem for spatial averages of the nonlinear SHE with drift, achieving a convergence rate proportional to R^(-1/2).
• It employs advanced Malliavin calculus, second-order Poincaré inequalities, and novel heat kernel product estimates to control nonlinear drift effects.
• The study extends Gaussian fluctuation phenomena to SPDEs with drift by proving ergodicity and a functional CLT with an explicit covariance structure.

Gaussian Fluctuations in the Nonlinear Stochastic Heat Equation with Drift

Problem Setting and Context

This paper addresses the quantitative central limit theorem (QCLT) for the spatial average of solutions to the one-dimensional nonlinear stochastic heat equation (SHE) with drift, perturbed by space-time Gaussian white noise. The equation of interest is

$$\frac{\partial u}{\partial t} (t,x) = \frac{1}{2}\frac{\partial^2 u}{\partial x^2} (t,x) + b(u(t,x)) + \sigma(u(t,x)) \dot{W}(t,x),$$

with constant initial condition and under the assumption that the drift $b$ and diffusion $\sigma$ are twice differentiable with bounded first and second derivatives. The focus is on the asymptotic distribution, as $R \to \infty$, of the (centered) spatial integral

$$F_R(t) = \int_{|x|\leq R} \left( u(t,x) - \mathbb{E}[u(t,x)] \right) dx.$$

The role of drift terms in fluctuation phenomena for stochastic PDEs has been notably under-investigated. Prior QCLT results largely concern SHEs and stochastic wave equations without drift or under limited forms of nonlinearity, typically focusing on the influence of the noise's spatial/temporal structure and omitting drift-driven terms. The present work fills this theoretical gap for the nonlinear SHE with drift in dimension one and provides both moment estimates and explicit convergence rates.

Methods: Malliavin Calculus, Poincaré Inequalities, and Heat Kernel Analysis

The core methodology combines advanced stochastic analysis tools:

• Malliavin Calculus: The solution $u(t,x)$, viewed as an adapted random field, is analyzed through its first and second Malliavin derivatives. Applying chain and commutation rules as well as ergodic theorems for random fields, the Malliavin derivative regularity is exploited.
• Second-Order Poincaré Inequality (Vidotto 2020): The quantitative normal approximation leverages the refined Poincaré inequality, bounding the total variation distance between the normalized spatial average and the Gaussian distribution by an explicit function of the second Malliavin derivative.
• Novel Heat Kernel Product Estimates: A key technical contribution is a new upper estimate for the product of two heat kernels, critical for analyzing second-order Malliavin derivative terms in the nonlinear setting. The bound

$$\| D^2_{(r,z),(\theta,w)}u(t,x)\|_p \lesssim (1 + |r-\theta|^{-1/2}) G_{8T}(z-w) (G_{t-r}(x-z) + G_{t-\theta}(x-w))$$

is established, overcoming obstacles arising from nonlinear terms and the lack of explicit solution formulas in the presence of drift.

For the CLT rate bound, the solution $F_R(t)$ is represented via the divergence operator as $F_R(t) = \delta(-DL^{-1}F_R(t))$, and the variance controlling the normal approximation is expressed in terms of contraction operators on the second Malliavin derivative.

Main Results

Ergodicity and Scaling

Theorem 1 shows that (i) the solution is spatially ergodic and (ii) the variance of the spatial average grows linearly with the integration radius; that is, for fixed $t$, $\operatorname{Var}(F_R(t)) \sim R$ as $R\to\infty$. The proof relies on boundedness and moment estimates for the Malliavin derivatives, implying effective decorrelation at large spatial scales.

Quantitative Central Limit Theorem

The central result is that, after normalization,

$$d_{TV}\left( \frac{F_R(t)}{\sqrt{\operatorname{Var}(F_R(t))}},Z\right) \leq C R^{-1/2},$$

with $Z$ standard normal and $C$ depending on equation parameters and time $t$. This parallels prior rates found in the setting of SHEs without drift and elucidates the robustness of the fluctuation exponent to the inclusion of a drift. The proof uses the second-order Poincaré inequality and requires intricate control on second-order Malliavin derivatives, for which the novel heat kernel product estimates are critical.

Functional Central Limit Theorem (FCLT)

A functional version is also proven: the renormalized processes $\{R^{-1/2} F_R(t)\}_{t \geq 0}$ converge in law (in $C([0,\infty))$) to a centered Gaussian process with explicit covariance structure derived from the $u$ process. The process exhibits Hölder-continuous sample paths of any order $\gamma < 1/2$.

Novelty and Features

• Inclusion of Drift: Previous QCLT and FCLT results for stochastic PDEs with spatial averaging are extended to equations with nonlinear drift. This removes a significant restriction of the earlier literature.
• Sharp Rate of Convergence: The total variation distance decay rate $R^{-1/2}$ is obtained, matching the optimal rate in white-noise-driven settings.
• Technical Advances: The handling of second Malliavin derivatives in nonlinear equations with drift required new kernel product bounds, which may facilitate further advances in analyzing fluctuations of nonlinear SPDEs.
• Explicit Functional Results: Tightness, finite-dimensional convergence, and path regularity of the spatial averages are established, providing a full invariance principle.

Implications and Future Directions

The work demonstrates the persistence of Gaussian fluctuation phenomena for spatial averages in nonlinear stochastic heat equations—even when nonlinear drift is present. This underscores the universality of CLT scaling in spatially extended SPDEs subject to homogeneous noise and robust nonlinearities. The techniques (especially the Malliavin derivative estimates and heat kernel analysis) are likely adaptable to higher-dimensional settings, equations with colored or rough noise, and other SPDEs (e.g., stochastic wave or reaction-diffusion equations). Such extensions are natural avenues for ongoing research.

Additionally, the methods employed may be applicable to the study of non-Gaussian or heavy-tailed noise, spatial ergodicity in more general geometries, and large deviations of spatial averages. The characterization of the rate and underlying structure of spatial decorrelation mechanisms in nonlinear SPDEs has direct theoretical implications and may inform simulation-based approaches to rare event estimation in random media or continuum models for physics, biology, and finance.

Conclusion

This paper delivers a quantitative central and functional limit theory for the spatial average of the nonlinear stochastic heat equation with drift under space-time white noise, establishing explicit convergence rates and regularity of the limit law. The analysis leverages advanced Malliavin calculus, a refined Gaussian Poincaré inequality, and novel heat kernel estimates. The results confirm and extend the universality of Gaussian fluctuation scalings in the presence of nonlinear drift and provide technical tools for future investigations into fluctuation phenomena in SPDEs with general nonlinearities.