Special Session 191: Stochastic Dynamical Systems Under Levy Noise: Theory and Applications

Critical Transitions in Multi-Particle Systems: A Generative Control Framework via Onsager Machlup Functionals

Jianyu Chen Nanyang Technological University Singapore Co-Author(s):    Jianyu Chen, Ting Gao, Jinqiao Duan

Abstract: This paper studies the most probable transition paths of high dimensional stochastic interacting particle systems and their mean field limits. Since these paths are difficult to compute directly, we reformulate the problem as a mean field optimal control problem based on the Onsager Machlup action functional. Using the stochastic Pontryagin maximum principle, we prove existence and uniqueness of the solution and derive a coupled system for the control variables. We also show that the Hamiltonian extremum conditions are equivalent to the conditions from the Pontryagin maximum principle. This equivalence gives an indirect characterization of the most probable transition paths and shows their correspondence with the paths of the mean field limit system.

How Mathematical Structures Emerge from Uncertainties: Dynamics, Geometry, and Topology

Ting Gao Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: Uncertainty is inherent in data generation processes, whether arising from stochastic dynamics, limited samples, or complex multi-scale interactions. Understanding how structured patterns emerge from such uncertainties is a central challenge in generative modeling. This report explores this question through the lens of dynamics, geometry, and topology, with a particular focus on early warning prediction. We investigate the mechanisms underlying critical transitions in generative models, including mode collapse and vector field splitting, which manifest as topological changes across scales. We introduce entropy-based indicators defined in the space of probability measures to assess and anticipate such transitions.

A kernel method for the learning of Wasserstein geometric flows

Jianyu Hu Nanyang Technological University Singapore Co-Author(s):

Abstract: In this talk, we address the inverse problem of simultaneously recovering free energy defined on the density manifold from discretized observations of the density flow, which generated by Wasserstein gradient or Hamiltonian flow. We formulate the problem as an optimization task that minimizes a loss function specifically designed to enforce the underlying variational structure of Wasserstein flows, ensuring consistency with the geometric properties of the density manifold. Our framework employs a kernel-based operator approach using the associated Reproducing Kernel Hilbert Space (RKHS), which provides a closed-form representation of the unknown components. Furthermore, a comprehensive error analysis is conducted, providing convergence rates under adaptive regularization parameters as the temporal and spatial discretization mesh sizes tend to zero. Finally, a stability analysis is presented to bridge the gap between discrete trajectory data and continuous-time flow dynamics for the Wasserstein Hamiltonian flow.

Nonlocal Kramers-Moyal formulas and data-driven discovery of stochastic dynamical systems with multiplicative Levy noise

Yang Li Nanjing University of Science and Technology Peoples Rep of China Co-Author(s):    Jinqiao Duan

Abstract: Traditional data-driven methods, effective for deterministic systems or stochastic differential equations (SDEs) with Gaussian noise, fail to handle the discontinuous sample paths and heavy-tailed fluctuations characteristic of L\`evy processes, particularly when the noise is state-dependent. To bridge this gap, we establish nonlocal Kramers-Moyal formulas, rigorously generalizing the classical Kramers-Moyal relations to SDEs with multiplicative L\`evy noise. These formulas provide a direct link between short-time transition probability densities (or sample path statistics) and the underlying SDE coefficients: the drift vector, diffusion matrix, L\`evy jump measure kernel, and L\`evy noise intensity functions. Leveraging these theoretical foundations, we develop novel data-driven algorithms capable of simultaneously identifying all governing components from data and establish convergence results and error analysis for the algorithms. We validate the framework through extensive numerical experiments on prototypical systems. This work provides a principled and practical toolbox for discovering interpretable SDE models governing complex systems influenced by discontinuous, heavy-tailed, state-dependent fluctuations, with broad applicability in climate science, neuroscience, epidemiology, finance, and biological physics.

Learning Stochastic Dynamical Systems via An Energetic Variational Approach

Yubin Lu South China University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present our recent work on learning stochastic dynamical systems through an energetic variational approach, and discuss its connections with existing methods. I will then present a series of numerical experiments to demonstrate the effectiveness and robustness of the proposed framework, particularly in the presence of noisy and partial observations.

Local and Global Solutions for Stochastic Transport-Type Equations with Nonlocal Noise

Yingting Miao Xi'an Jiaotong-Liverpool University Peoples Rep of China Co-Author(s):

Abstract: This presentation discusses results on stochastic transport-type equations driven by genuinely mixed multiplicative noise. The noise comprises a continuous component (interpreted in both the Stratonovich and It\^{o} senses) and a discontinuous component (interpreted in the Marcus sense). The Stratonovich and Marcus noise amplitudes are given by (nonlocal) pseudo-differential operators, which include the classical transport operator as a special case. Within this setting, we establish the existence, uniqueness, and a blow-up criterion for pathwise classical solutions. We further prove that sufficiently fast-growing noise ensures global regularity.

Anomalous branching processes: probabilistic behavior and connection to nonlinear time-fractional PDEs

Nicolas Privault Nanyang Technological University Singapore Co-Author(s):    Gerardo Perez-Suarez and Nicolas Privault

Abstract: We present a class of non-Markovian branching processes, called anomalous branching processes, and study their probabilistic properties, such as the behavior of the moments of the number of particles alive at a time t on a given set. Those processes are then used to derive a probabilistic representation for the solutions of time-fractional F-KPP equations. By exploiting this connection, we also derive estimates for the probability tail of the position of the rightmost particle in the one-dimensional case.

Identifying and Predicting Critical Transitions in a Cdc2-Cyclin B/Weel System

Hui Wang Zhengzhou University Peoples Rep of China Co-Author(s):

Abstract: Critical transitions in intracellular regulatory networks, driven by fluctuations in protein concentrations, are fundamental to cellular decision-making and disease-related state transition. We investigate the stochastic dynamics of the Cdc2-Cyclin B/Wee1 regulatory module under Gaussian noise, focusing on how variations in feedback strength modulate system stability and transition behavior. By analyzing time-series data of protein concentrations, we evaluate a set of statistical, informational, and dynamical indicators to identify early signatures of critical transitions. Feature selection techniques are employed to determine the most informative indicators, which are subsequently incorporated into a deep learning framework based on a multilayer perceptron (MLP) regression model to predict the future evolution of Cdc2-Cyclin B concentrations and provide early warning of impending transitions.

Modeling Epidemic Dynamics: A Stochastic SIR Framework with Tempered Stable Distributions

YUNFEI XIA Harbin Engineering University Peoples Rep of China Co-Author(s):    Yang Jia; Ling Xue

Abstract: We study a stochastic SIR model driven by multivariate tempered stable (TS) to capture heavy-tailed, cross-compartment shocks. The noise has a tail index $\alpha \in(0,1)$ with exponential tempering, accommodating rare, large jumps while preserving finite second moments and flexible dependence. Within a spectral-radial specification, finite spectral measure on the unit sphere coupled with a prescribed radial density, we characterize the infinite L\`evy measure and the resulting dynamics. We establish a sharp threshold separating extinction from persistence: below the threshold, the epidemic dies out; above it, the process admits a unique ergodic stationary distribution. Monte Carlo experiments reproduce jump-triggered flare-ups and abrupt regime shifts. Overall, our results show that epidemic persistence is governed not only by variance, but critically by tail shape and the magnitude of large jumps, underscoring the relevance of TS noise for data with sudden outbreaks and correlated shocks.

Convergence of One--Dimensional Ising--Kac--Kawasaki Dynamics to Stochastic Cahn--Hilliard Equations

Qi Zhang Beijing Institute of Mathematical Sciences and Applications Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider the scaling limit of the one-dimensional lattice Ising--Kac--Kawasaki dynamics under conservative Kawasaki exchange rate. For the Kac coarse-grained field \(X_\gamma\), we derive a martingale formulation with a discrete conservative drift and a Dynkin martingale. The nonlinear drift is identified by a conservative multiscale replacement scheme based on one-block/two-block estimates, yielding a cubic conservative term in the macroscopic equation. For the stochastic part, we compute the predictable quadratic variation, and obtain a divergence-form Gaussian noise. As a consequence, \(X_\gamma\) converges to a one-dimensional stochastic Cahn--Hilliard equation with conserved noise.