Langevin Diffusion

• Langevin diffusion is a family of continuous-time stochastic processes defined by SDEs that engineer drift and noise to achieve a prescribed invariant measure.
• It generalizes to non-Euclidean domains, heterogeneous media, and constrained environments, facilitating advanced applications in sampling and statistical inference.
• Innovations such as state-dependent diffusivity, memory kernels, and underdamped formulations enable robust mixing, efficient MCMC proposals, and generative modeling techniques.

Langevin diffusion refers to a family of continuous-time stochastic processes defined by stochastic differential equations (SDEs), wherein the drift and noise structure are engineered so that a prescribed invariant measure is realized at long times. Originally emerging in the context of statistical physics as a description of Brownian motion and fluctuation–dissipation phenomena, Langevin diffusions have subsequently become foundational in probability theory, sampling algorithms, statistical inference, and mathematical modeling of constrained and out-of-equilibrium systems. The formalism generalizes naturally to non-Euclidean domains (e.g., manifolds), non-equilibrium systems (with non-trivial memory kernels), heterogeneous and constrained environments, and high-dimensional or structured-data applications.

1. Mathematical Formulation of the Langevin Diffusion

A canonical overdamped Langevin diffusion in Euclidean space is governed by the SDE

$$dX_t = \frac{1}{2} \nabla \log \pi(X_t)\,dt + dW_t,$$

where $\pi(x)$ is a target invariant density (with respect to Lebesgue measure) and $W_t$ is standard Brownian motion. The diffusion is constructed so that the process $X_t$ has $\pi$ as its stationary distribution. The drift term $(1/2)\nabla \log \pi$ ensures detailed balance with respect to $\pi$ and is formally equivalent to a gradient ascent on the log-density regularized by noise.

Generalizations include Langevin diffusions with state-dependent preconditioning (diffusivity) matrices $A(x)$, as in

$$dX_t = \frac{1}{2} A(x) \nabla \log \pi(X_t)\,dt + \sqrt{A(x)}\,dW_t,$$

requiring an additional correction term when $A$ varies in position to ensure $\pi$ is stationary (Xifara et al., 2013). Similarly, on Riemannian manifolds, the diffusion is described intrinsically by

$$dX_t = -\frac{1}{2} \mathrm{grad}_g \phi(X_t)\,dt + dB^M_t,$$

where $\phi$ is a smooth potential, $dB^M_t$ is intrinsic Brownian motion, and the invariant measure is $d\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g$ (Bharath et al., 2023).

The underdamped (kinetic) Langevin diffusion includes position $x_t$ and velocity $v_t$ variables, typically written as

$$dx_t = v_t dt, \qquad dv_t = -\gamma v_t dt + \nabla \log \pi(x_t) dt + \sqrt{2\gamma}\,dW_t,$$

and is used extensively both in molecular simulation and in modern score-based generative models (Geffner et al., 2022, Cornet et al., 4 Jul 2025).

A key structural property is that the Fokker–Planck equation for the process yields (by construction) a stationary solution proportional to $\pi$.

2. Langevin Diffusion in Structured and Constrained Environments

Langevin diffusion admits crucial extensions to non-Euclidean domains, nonhomogeneous media, and heterogeneous or constrained systems.

• Langevin Diffusions on Manifolds: On compact Riemannian manifolds the process is intrinsically defined, driven by the Riemannian gradient and canonical Brownian noise (Bharath et al., 2023). For coordinates $X$,

$$dX(t) = -\frac{1}{2} \mathrm{grad}_g \phi(X(t)) dt + dB^M(t),$$

inducing invariant measure $d\mu_\phi \propto e^{-\phi} dvol_g$. First-order weak error bounds for discretized samplers match the classical Euclidean case, including when using retraction maps instead of exact geodesics.

• Position-Dependent Diffusivity (Heterogeneous Media): In nonhomogeneous environments, e.g., where $D(x)$ changes abruptly (two-phase systems), the SDE is

$dX(t) = \sqrt{2D(X_t)}\,dB(t).$

The value of the stochastic integral is ambiguous unless a discretization/interpolation convention is chosen: Itô ($\alpha=0$), Stratonovich ($\alpha=1/2$), or Hänggi–Klimontovich ($\alpha=1$). The correct interpretation must be inferred from physical or experimental input and directly impacts observables such as the probability density, mean, and mean-square displacement (Pacheco-Pozo et al., 18 Mar 2024).

• Anomalous Diffusion and Memory Kernels: Generalized Langevin equations with power-law memory kernels $\tilde\gamma(t) \sim t^{-\beta}$ and colored noise reproduce subdiffusive or superdiffusive transport, as in single-file diffusion,

$$dx(t)/dt = v(t),\quad dv(t)/dt = -\int_0^t \tilde\gamma(t-t')v(t')\,dt' + \xi(t),$$

where the memory kernel imparts crossover dynamics from ballistic to subdiffusive scaling in mean-square displacement (0810.5210, Pereira-Alves et al., 2022, Kimura et al., 2023).

• Systems with Fluctuating Diffusivity: For situations with time-dependent $D(t)$ (e.g., reflecting conformational fluctuations in proteins), the effective long-time diffusion coefficient can exhibit persistent dependence on initial conditions if the modulating process is non-Markovian, especially with power-law memory (Miyaguchi et al., 2016, Kimura et al., 2023).

3. Invariant Measures, Stationarity, and Ergodicity

The invariant measure of a Langevin process is determined by the drift structure and the noise covariance. In the basic overdamped case in $\mathbb{R}^d$,

$\pi(x) \propto \exp\left(2 \int b(x)\,dx \right)$

if the drift is $b(x) = (1/2)\nabla \log \pi(x)$. On manifolds, the invariant density is always with respect to the Riemannian volume form. Position-dependent diffusions on manifolds require careful attention to the distinction between invariant measures with respect to the Hausdorff measure and those with respect to Lebesgue/Darboux coordinates (Xifara et al., 2013).

A remarkable feature of many Langevin diffusions is exponential erosion of statistical dependence along the chain under strong log-concavity: the mutual information between initial and later samples decays as $\sim \exp(-2\alpha t)$ in continuous time (with $\alpha$ the strong convexity parameter), ensuring both fast mixing and approximate independence of distant samples (Liang et al., 26 Feb 2024).

4. Generalized Fluctuation–Dissipation and Linear Response

The formalism unifies equilibrium fluctuation–dissipation and non-equilibrium linear response:

• In generalized Langevin systems with memory kernel $\tilde\gamma(t)$, the noise covariance is constrained by the generalized fluctuation–dissipation theorem:

$$\langle \xi(t)\xi(t')\rangle = k_B T\, \tilde\gamma(|t-t'|).$$

This ensures the correct equilibrium distribution in the stationary state (0810.5210, Martínez-Mesa et al., 2018). For Langevin processes driven by non-Markovian kernels, linear response coefficients derived via Green–Kubo relations yield the complex mobility, and thus the system's response to oscillatory perturbations across all dynamic regimes.

• In molecular diffusion models subjected to time-dependent fields or periodic optical lattices, the Langevin approach combined with the system–bath formalism allows the explicit calculation and control of effective damping and noise via external tuning of spectral densities (Martínez-Mesa et al., 2018).

5. Sampling, Inference, and Generative Modeling

Langevin diffusions underpin numerous algorithmic advances in high-dimensional inference and generative modeling:

• MCMC and Langevin Proposals: The Metropolis-adjusted Langevin algorithm (MALA) and its manifold generalizations exploit Langevin SDEs to propose updates with high acceptance rates. Correct specification of the drift and correction terms ensures invariance of the target density (Xifara et al., 2013). Newer position-dependent MALA proposals are computationally more efficient and correct earlier errors related to the invariant measure.
• Replica Exchange and Multimodal Sampling: For targets $\pi(x)$ with multiple isolated modes, simple Langevin dynamics mixes poorly. Replica Exchange Langevin Diffusion (ReLD) and its multiple-replica version (mReLD) introduce auxiliary high-temperature processes and Metropolis-exchange steps to accelerate mixing time, obtaining spectral gaps that do not deteriorate with increasing concentration or separation of the modes (Dong et al., 2020, Chen et al., 2020). Theoretical work quantifies how careful temperature ladder selection and exchange rates yield constant spectral gaps, even for sharply multimodal systems.
• Variational Inference and Diffusion Models: Recent work has unified a range of MCMC-based and score-based VI methods via Langevin dynamics. In particular, underdamped Langevin SDEs and their time-reversed analogs are combined with neural score networks to match complex target distributions and optimize the evidence lower bound (ELBO), yielding state-of-the-art results on a range of practical inference problems (Geffner et al., 2022).
• Generative Modeling on Manifolds and Structured Domains: In crystalline materials generation, KLDM introduces a velocity-coupled Langevin SDE on the hypertorus, leveraging the auxiliary velocity’s Euclidean structure to guarantee manifold consistency and periodic translation invariance (Cornet et al., 4 Jul 2025).
• Score-based Diffusion and Schrödinger Bridges: At small time step, the Langevin diffusion provides a finite-temperature approximation to the Schrödinger bridge with the same marginal. The leading-order deviation from the identity transport (in the regularized entropic OT map) is proportional to the score function $\nabla \log \rho$, and the associated Markov operators inherit an approximate semigroup property, with generator the standard Langevin operator (Agarwal et al., 12 May 2025).

6. Physical Realizations, Constraints, and universality

• Single-File Diffusion: In 1D systems where particles cannot pass each other, the Langevin formulation includes a power-law nonlocal memory kernel (e.g., $\tilde\gamma(t) \sim t^{-3/2}$), enforcing subdiffusive scaling and capturing three qualitatively distinct dynamical regimes—ballistic, normal diffusive, and long-time subdiffusive (0810.5210).
• Holographic and Anisotropic Plasmas: In gauge/gravity duality, the process of heavy quark thermalization and momentum broadening is encoded in holographic Langevin coefficients computed from trailing strings in a black-hole geometry (Gursoy et al., 2010, Zhou et al., 3 Sep 2024). These coefficients exhibit universal relations (e.g. longitudinal coefficient larger than transverse in isotropic backgrounds), but can be violated in anisotropic environments or in the presence of strong external fields, with direct phenomenological consequences for quark–gluon plasma observables (Giataganas et al., 2013, Zhou et al., 3 Sep 2024).
• NMR and Fractional Langevin Models: Anomalous spin diffusion in confined geometries, as probed in NMR, is modeled by fractional Langevin equations with power-law kernels leading to non-exponential (Mittag–Leffler) decay in relaxation and non-trivial scaling in the echo attenuation signal (Pereira-Alves et al., 2022).
• Complexified and AI-Learned Langevin Dynamics: For systems with a sign problem (e.g., field theory at finite chemical potential), complex Langevin dynamics is employed to sample from distributions not directly addressed by importance sampling. Recent work leverages diffusion models (generative AI) to learn the complexified stationary distribution produced by the Langevin process, using learned scores to reconstruct or analyze the underlying effective action (Habibi et al., 2 Dec 2024).

7. Summary Table: Core Langevin Diffusion Variants and Features

Domain / Context	Key SDE Structure	Symmetry/Feature
Euclidean (flat)	$dX_t = (1/2) \nabla \log \pi(X_t)\,dt + dW_t$	$\pi$ is invariant measure w.r.t. Lebesgue
Riemannian Manifold	$dX_t = -(1/2)\mathrm{grad}_g \phi(X_t)dt + dB^M_t$	Invariant: $e^{-\phi}dvol_g$
Single-File	$$dv/dt = -\int_0^t \tilde\gamma(t-t')v(t')dt' + \xi(t)$$	Memory kernel, subdiffusion
Position-dependent D	$dX_t = \sqrt{2D(X_t)} dB_t$ (interpretation $\alpha$-dependent)	Martingale, biased, or continuous
Kinetic/Lifted	$dx_t = v_tdt$, $dv_t = -\lambda v_tdt + \sqrt{2\lambda}dW_t$	Second-order, momentum-enriched
Anomalous/Memory	$$dv/dt = -\int \tilde\gamma(\cdot) v(\cdot) + \xi(\cdot)$$	Non-Markovian, subdiffusion
Holographic	Fluctuation equations for trailing string	Multiple diffusion directions

• "Langevin formulation for single-file diffusion" (0810.5210)
• "Langevin diffusion of heavy quarks in non-conformal holographic backgrounds" (Gursoy et al., 2010)
• "Langevin diffusions and the Metropolis-adjusted Langevin algorithm" (Xifara et al., 2013)
• "Universal Properties of the Langevin Diffusion Coefficients" (Giataganas et al., 2013)
• "Anomalous diffusion in nonhomogeneous media: Time-subordinated Langevin equation approach" (Srokowski, 2014)
• "Langevin equation with fluctuating diffusivity: a two-state model" (Miyaguchi et al., 2016)
• "The Langevin diffusion as a continuous-time model of animal movement and habitat selection" (Michelot et al., 2018)
• "Effective Langevin Equation Approach to the Molecular Diffusion on Optical Lattices" (Martínez-Mesa et al., 2018)
• "Spectral Gap of Replica Exchange Langevin Diffusion on Mixture Distributions" (Dong et al., 2020)
• "Accelerating Nonconvex Learning via Replica Exchange Langevin Diffusion" (Chen et al., 2020)
• "NMR diffusion in restricted environment approached by a fractional Langevin model" (Pereira-Alves et al., 2022)
• "Langevin Diffusion Variational Inference" (Geffner et al., 2022)
• "Non-Markovian effects of conformational fluctuations on the global diffusivity in Langevin equation with fluctuating diffusivity" (Kimura et al., 2023)
• "Sampling and estimation on manifolds using the Langevin diffusion" (Bharath et al., 2023)
• "Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of $Φ$-Mutual Information" (Liang et al., 26 Feb 2024)
• "Langevin equation in heterogeneous landscapes: how to choose the interpretation" (Pacheco-Pozo et al., 18 Mar 2024)
• "Aspects of holographic Langevin diffusion in the presence of anisotropic magnetic field" (Zhou et al., 3 Sep 2024)
• "Diffusion models learn distributions generated by complex Langevin dynamics" (Habibi et al., 2 Dec 2024)
• "Langevin Diffusion Approximation to Same Marginal Schrödinger Bridge" (Agarwal et al., 12 May 2025)
• "Kinetic Langevin Diffusion for Crystalline Materials Generation" (Cornet et al., 4 Jul 2025)