probabilistic coupling

This work investigates how information influences the propagation of chaos in stochastic interacting particle systems under the mean-field limit. The problem centers on quantifying and analyzing chaos propagation in path space for systems governed by stochastic differential equations. We propose the first framework integrating information-theoretic principles into path-space chaos analysis: modeling chaos propagation as an information divergence problem over the space of sample paths, then leveraging the data processing inequality to reduce it to estimating drift-term discrepancies—thereby circumventing conventional techniques based on hypocoercivity or pseudo-inverses of diffusion matrices. Our approach unifies tools from stochastic analysis, McKean–Vlasov theory, and path-space probability measures. Validated on canonical second-order systems, the method substantially simplifies convergence proofs, enhances numerical stability, and improves generality across diverse mean-field models.

This work studies optimization over probability distributions in the 2-Wasserstein space, focusing on convergence guarantees of empirical measures—induced by finite particle systems—to the optimal distribution. We propose the Virtual Particle Stochastic Approximation (VP-SA) algorithm, the first systematic integration of the virtual particle paradigm into mean-field optimization, which ensures strong convergence of i.i.d. particle empirical measures to the optimal distribution without relying on propagation-of-chaos analysis. VP-SA unifies concepts from Wasserstein gradient flows, Stein variational gradient descent, and stochastic optimization theory, achieving convergence at the standard SGD rate while preserving the i.i.d. property of output samples. Compared to prior approaches, our theoretical assumptions are significantly weakened: convergence conditions match those of the infinite-particle limit. This yields a more concise and practically applicable theoretical and algorithmic foundation for distributional optimization.

This study investigates the stationary distribution and stability of stochastic differential equation systems with multimodal uncertain parameters exhibiting superposition effects, using the nonlinear Rosenzweig–MacArthur predator–prey model as a case study. For the first time, multimodal mixture-distributed parameters are incorporated into the stationary analysis of stochastic dynamical systems. System stability is quantified through the eigenvalue distribution of the Jacobian matrix, and posterior stationary density estimates are obtained via the Monte Carlo method proposed by Hoegele (2026). The results reveal that under multimodal parameter uncertainty, the system exhibits a multimodal stationary distribution, accurately delineating regions of stability. This demonstrates the effectiveness and novelty of the proposed framework for uncertainty quantification in complex ecological dynamics.

This work addresses the problem of identifying the optimal coupling between input and output distributions generated by a causal dynamical system, subject to both a prescribed causal temporal structure and given Gaussian marginals. The authors formulate this as a Schrödinger bridge problem with explicit causality constraints, minimizing the Kullback–Leibler divergence to a prior coupling under a time-varying quadratic cost function. Their key contribution is the derivation of a closed-form Sinkhorn-type iterative algorithm for the Gaussian setting, enabling efficient and analytically tractable solutions to the causal optimal transport problem. The proposed method rigorously preserves causality and temporal consistency while offering a novel theoretical framework and computational paradigm for distributional data-driven system identification.

This paper investigates the asymptotic behavior of mean-field Langevin dynamics and their associated particle systems under a functional convexity assumption on the energy. We address two central problems: (i) quantifying the convergence rate of marginal distributions to the unique invariant measure, and (ii) establishing time-uniform propagation of chaos. To this end, we develop a unified analytical framework integrating optimal transport theory, entropy estimates, and probabilistic metric analysis. Our main contribution is the first derivation of *time-uniform* quantitative propagation-of-chaos bounds—both in the $L^2$-Wasserstein distance and relative entropy—where the error bounds depend neither on the initial configuration nor on time horizon. Furthermore, we obtain explicit $L^p$-convergence rates. Crucially, these results circumvent the classical requirement of strong exponential convergence assumptions tied to restrictive initial conditions. The analysis provides a rigorous foundation for large-system limits in nonequilibrium statistical physics and for sampling-based optimization algorithms in machine learning.

This work addresses the challenge of unstable training in reinforcement learning within chaotic dynamical systems, where exponential sensitivity to initial conditions induces high variance in bootstrapped targets and ill-conditioned gradient updates. To mitigate this, the authors propose a distributional reinforcement learning approach that models the return distribution under the 1-Wasserstein metric. Leveraging the observation that the evolution of return distributions is smoother than individual trajectories, they formulate a well-conditioned Bellman optimization objective. The study provides the first theoretical insight from the perspective of measure evolution, demonstrating that return distributions exhibit more regular dynamical structure even in chaotic regimes. This analysis elucidates the geometric properties of the resulting objective function and offers principled justification for enhanced learning stability in chaotic environments.

This study addresses the limitations of traditional state-space models, which rely on predefined nonlinear dynamics and struggle with theoretically under-specified complex systems, as well as the high computational cost of Bayesian inference in Gaussian process state-space models for moderately long sequences. To overcome these challenges, the authors propose two enhanced Gibbs sampling strategies that substantially improve sampling efficiency and convergence reliability. By integrating confirmatory factor analysis to construct an identifiable and interpretable measurement structure, they develop a comprehensive framework for learning nonlinear latent dynamical systems. Simulation studies validate the accuracy of posterior inference, while two empirical applications demonstrate the method’s practical utility and interpretability. An open-source implementation is provided, offering researchers an efficient and feasible workflow for empirical analysis.

This work addresses the challenges of model inaccuracy and ambiguous state distributions in nonlinear systems by proposing a distributionally robust optimization-based chance-constrained control framework. The approach constructs an ambiguity set using relative entropy constraints and derives an upper bound on risk expectations via the variational representation of the exponential integral. It further integrates nonlinear covariance propagation with adaptive determination of the ambiguity set radius based on second-order dynamic truncation error. Notably, the method recovers nominal risk in the zero-divergence limit, thereby overcoming the restrictive assumptions of Gaussianity prevalent in conventional approaches. Validation on spacecraft stochastic guidance tasks demonstrates that the proposed framework effectively enforces probabilistic safety constraints under distributional uncertainty.

Chaotic systems are highly sensitive to modeling errors, making it challenging to simultaneously achieve accuracy in local dynamics and long-term statistical fidelity. To address this, this work proposes a novel approach that constructs a local cover of the chaotic attractor in phase space and jointly optimizes both the Jacobian accuracy of the surrogate model and its long-term statistical behavior. The method uniquely integrates local pushforward distribution matching with Jacobian fidelity, thereby unifying local and global modeling paradigms. Training employs a loss function based on distribution matching, utilizing the Maximum Mean Discrepancy (MMD) metric. Experimental results demonstrate that the proposed method substantially improves Jacobian accuracy while achieving state-of-the-art performance in long-term statistical properties.

This work addresses the challenge of efficiently sampling from Gibbs distributions in complex energy landscapes characterized by barriers or metastable states. The authors propose a hybrid stochastic dynamics framework that employs two distinct sampling dynamics in different regions of the state space, coupled at their interface through a natural transmission condition that preserves the target distribution. By introducing a regularization mechanism, they establish—for the first time—the exponential convergence rate of this hybrid dynamics. In radially symmetric potentials, the method significantly reduces the mean escape time compared to conventional approaches. Both theoretical analysis and numerical experiments demonstrate that the proposed scheme offers marked improvements over traditional sampling strategies in terms of convergence speed and the ability to overcome metastability.