Learning Equilibrium Fluctuation Expansions from Overdamped Langevin Dynamics

Abstract: We study higher-order small-noise fluctuation expansions for the overdamped Langevin dynamics in a quartic double-well potential. Assuming that the initial data admits a suitable expansion structure, we obtain a strong dynamical expansion of the trajectories, as well as an expansion of the laws with respect to smooth observables. We then investigate the long-time behavior of the expansion coefficients. In the scalar case $d=1$, each coefficient converges exponentially fast to a finite limit as $t\to\infty$. In contrast, for $d\ge 2$, the fluctuation expansion coefficients reflect the degeneracy of the manifold of minima, which in general prevents the existence of a finite long-time limit. Furthermore, by combining a multi-level induction with combinatorial arguments, we derive a recursive formula for the fluctuation expansion coefficients. This recursion shows that the long-time limits of these dynamical expansion coefficients coincide with those arising from the corresponding equilibrium expansions.

• The paper establishes a recursive framework to extract higher-order fluctuation coefficients from Langevin trajectories without explicit knowledge of equilibrium measures.
• It demonstrates that in nondegenerate scalar cases, the dynamical coefficients converge exponentially to their equilibrium counterparts, while degenerate vector cases show unbounded behavior.
• The work bridges SDE perturbation theory and Laplace asymptotics, offering insights for improved uncertainty quantification and advanced stochastic sampling algorithms.

Higher-Order Equilibrium Fluctuation Expansions via Overdamped Langevin Trajectories

Introduction and Motivation

This work addresses the problem of extracting detailed fluctuation information (beyond CLT order) about equilibrium statistical quantities associated with ergodic stochastic differential equations, via direct analysis of trajectories. The prototype is the overdamped Langevin equation in a quartic double-well potential,

$$d X_\varepsilon(t) = -\nabla V(X_\varepsilon(t))\,dt + \sqrt{2\varepsilon}\,dW_t$$

with equilibrium measure

$$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$

where $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$, and the goal is to derive systematically higher-order small-$\varepsilon$ asymptotic expansions for both the time-dependent law and time-averaged (equilibrium) observables, tracking terms well beyond the leading-order CLT scaling.

The central questions are: to what extent can these higher-order equilibrium fluctuation terms be “learned” dynamically from Langevin trajectories without explicit knowledge of $\mu_\varepsilon$? How does this behavior depend on the geometry of the potential (e.g., presence of degenerate minima)? What are the precise recursive structures satisfied by the coefficients of these expansions, and what is their long-time behavior?

Formal Fluctuation Expansions: Construction and Recursion

The analysis focuses on constructing an expansion of the form (for small $\varepsilon$),

$$X_\varepsilon(t) = \bar X_0(t) + \varepsilon^{1/2} \bar X_1(t) + \varepsilon\,\bar X_2(t) + \cdots + \varepsilon^{n/2} \bar X_n(t) + o(\varepsilon^{n/2}),$$

where each fluctuation coefficient $\bar X_k$ is a (generally non-Gaussian, possibly nonlinear) functional of Brownian motion, computable recursively by consistently matching powers in the SDE. The leading term $\bar X_0(t)$ is a deterministic ODE flow, the first-order $\bar X_1(t)$ is a linear SDE, and higher-order terms satisfy inhomogeneous linear equations with forcing built from lower-order coefficients.

Simultaneously, weak expansions of averaged observables are derived:

$$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$0

where the $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$1 are explicit linear combinations of functional derivatives of $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$2 acting on products and moments of the $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$3. These formulas extend Edgeworth expansions familiar from classical probability far beyond the CLT.

A precise, recursive combinatorial structure is established for the coefficients, expressible as nested expectations over functionals $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$4 involving products of the lower-order $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$5. The expansion is constructed with pathwise $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$6-control of the remainders, explicitly quantifying the uniformity in time on compacts and in the parameter $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$7.

Long-Time Asymptotics and Rigorous Identification with Equilibrium Expansions

A fundamental contribution is the rigorous study of the long-time limit $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$8 of the dynamical coefficients $$\mu_\varepsilon(dx) \propto \exp(-V(x)/\varepsilon)\,dx,$$9. The results exhibit a sharp dichotomy:

• Scalar Case ($V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$0): If the leading-order initial condition is nondegenerate, every coefficient $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$1 converges exponentially fast to a finite limit $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$2. These limits are shown, by matching explicit recursive relations, to coincide with the coefficients $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$3 in the equilibrium expansion of $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$4 constructed via Laplace's method. This identification justifies the interchange of small-noise and long-time limits in the scalar nondegenerate setting.
• Vector Case ($V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$5): In contrast, the presence of a degenerate manifold of minima (e.g., the sphere $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$6 for quartic potentials in $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$7) leads to non-converging behavior: certain fluctuation coefficients, specifically at second-order for suitable observables, exhibit unbounded or growing contributions as $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$8 due to tangential directions along the manifold. The loss of exponential contractivity (multiple zero Lyapunov exponents) breaks the direct correspondence with equilibrium fluctuations.

The technical analysis involves a multi-level nested induction over the recursion indices for the moments, with an explicit mapping to the structure of the Fokker–Planck equation for the invariant measure, and Laplace asymptotics at the minima of $V(x) = \frac{1}{4}|x|^4 - \frac{1}{2}|x|^2$9. This identifies the universality and limitations of the dynamical-to-equilibrium coefficient transfer principle.

Technical Results and Explicit Statements

Key quantitative achievements include:

• $\varepsilon$0-Estimates: For any finite time, polynomial moments of all expansion coefficients and remainders are uniformly bounded in $\varepsilon$1.
• Strong/Weak Asymptotic Expansions: The expansions hold strongly in $\varepsilon$2 and in the sense of distributions against a large class of observables.
• Exponential Convergence: In the scalar case, each $\varepsilon$3 converges to its limit at explicit exponential rates, with recursive bounds derived for all nested terms $\varepsilon$4.
• Non-Convergence Contradiction: In the degenerate vector case, it is shown constructively that the dynamical coefficients can grow without bound, and thus cannot capture the equilibrium expansion coefficients, regardless of initial data.
• Recursive Formula Equivalence: For symmetric double-well potentials, recursion satisfied by dynamical and equilibrium expansion coefficients is explicitly demonstrated to coincide for $\varepsilon$5 with even symmetry and nondegeneracy.

Theoretical and Practical Implications

The results yield a rigorous non-asymptotic understanding of how fine-grained equilibrium statistical information (e.g., bias and non-Gaussian corrections) can be learned from finite-time Langevin data—a principle relevant to probabilistic sampling, advanced SGLD algorithms, and high-accuracy uncertainty quantification in molecular and Bayesian computation contexts.

The work clarifies the critical role of the geometry of the underlying potential: well-separated (nondegenerate) minima permit uniform extraction of fluctuation data via time-dependent trajectories, while potentials with continuous manifolds of minima create persistent directions of slow mixing, radically altering the long-time fluctuation structure.

On the theoretical side, the analysis synthesizes SDE perturbation theory, combinatorial recursion, and Laplace/WKB asymptotics, suggesting that similar expansion machinery can be adapted to other classes of Markov processes, possibly in infinite dimensions or in metastability-dominated landscapes.

Generalizations and Connections

Although the methodology is carried out for the classical quartic double-well, the approach is extensible to more general smooth polynomial potentials, provided nondegeneracy and symmetry are appropriately handled at the minima. The recursion structures, while more complicated, can still be systematically derived and may differ depending on the local geometry.

Additionally, the analysis interfaces with large deviation and WKB expansions, providing rigorous polynomial-order refinement of logarithmic asymptotics à la Freidlin–Wentzell. There are also direct connections to the computation of Lyapunov exponents and synchronisation-by-noise phenomena in random dynamical systems, as higher-order fluctuations quantify sensitivity beyond linear response.

Conclusion

This paper provides a rigorous, systematized approach to learning higher-order equilibrium fluctuation expansions for overdamped Langevin dynamics, establishing when and how dynamical trajectory data encodes the full structure of equilibrium corrections and identifying sharp geometric obstructions to this transfer principle. The technical results bridge SDE asymptotics, spectral theory, and statistical computation, opening the door to more general applications in nonequilibrium limits and in the analysis of stochastic algorithms for high-dimensional sampling and optimization.

Reference: "Learning Equilibrium Fluctuation Expansions from Overdamped Langevin Dynamics" (2604.04100)