first-passage time analysis

This work investigates the reverse-time dynamics of generative diffusion models on low-dimensional Gaussian mixture manifolds embedded in high-dimensional ambient space, focusing on characterizing the onset times of two phase transitions: “species differentiation” and “collapse.” We derive an exact mutual information (free energy) formula based on a generalized linear model and perform asymptotic analysis in the joint large-dimension and exponential-sample-size limit. This yields the first closed-form analytical expressions for phase transition times as functions of key geometric parameters—including the ratio of manifold dimension to ambient dimension. Our results establish a quantitative link between manifold geometry and diffusion dynamics, revealing the critical mechanism underlying the implicit modeling capability of diffusion models on nontrivial manifolds. The theory provides a foundational understanding of training and sampling dynamics for diffusion models applied to real-world low-dimensional data.

This work addresses collective failure triggered by critical threshold crossings, focusing on optimizing first-passage time and mitigating catastrophic losses via synchronized resetting of search agents. We propose a novel “threshold-resetting” paradigm and develop an analytically tractable unified stochastic framework that captures event-driven, strongly coupled search dynamics. Contrary to conventional wisdom, we rigorously demonstrate—both theoretically and quantitatively—that resetting can *delay*, rather than accelerate, system failure, revealing a counterintuitive optimization mechanism. We extend the model to multi-degree-of-freedom systems and integrate ballistic searchers with joint cost-function optimization, combining stochastic process theory, first-passage analysis, and event-driven modeling. Our analysis uncovers rich non-monotonic optimal resetting behaviors and proves that threshold-resetting significantly reduces expected loss in failure-avoidance tasks. The framework provides both a new theoretical principle and a computationally feasible tool for robust control of critical systems. (149 words)

In large-scale real-time monitoring systems (e.g., communication networks), end-to-end delay autocorrelation significantly degrades Age of Information (AoI) performance—a critical limitation unaddressed under conventional single-server queueing assumptions. Method: We propose a general modeling framework that represents end-to-end delay as a nonnegative continuous-time virtual delay process, decoupling AoI analysis from restrictive queueing assumptions. For the first time in AoI theory, we employ Gaussian processes to rigorously characterize the impact of delay’s second-order statistics—particularly its autocorrelation structure—on AoI degradation. Contribution/Results: We derive closed-form analytical expressions for the transient AoI distribution and establish strict stochastic order relationships linking delay autocorrelation to AoI performance loss. Both theoretical analysis and numerical experiments confirm that strong delay autocorrelation substantially worsens AoI; we further quantify the sensitivity of AoI distribution and expected value to the delay covariance function. This work establishes a novel paradigm for AoI modeling and optimization in complex, correlated-network environments.

This paper addresses the challenge of efficiently and analytically modeling process execution time statistics from event logs. Methodologically, it introduces the first end-to-end analytical performance analysis framework based on semi-Markov processes: it directly infers execution time means and probability density functions (PDFs) from logs—bypassing simulation entirely. For discrete-time execution times, it employs exact convolution; for continuous-time cases, it approximates PDFs using Gaussian mixture models (GMMs), balancing accuracy, model compactness, and interpretability. Experiments show that the discrete-time approach achieves up to one order of magnitude speedup over simulation under small support sets, while GMM-based representation drastically reduces model size, enabling rapid what-if analysis. The core contribution is the first fully analytical, log-driven inference of semi-Markov performance models—eliminating reliance on traditional simulation-based approaches and establishing a new paradigm for scalable, interpretable process performance analysis.

This paper addresses the challenge of modeling population-level variability in replicated point processes. We propose a novel functional principal component analysis (fPCA) framework grounded in random measures and the cumulative mass function (CMF). Introducing the concept of “principal measures,” we establish the Karhunen–Loève expansion for random measures and derive a Mercer-type theorem for their covariance measures, enabling consistent parameter-rate estimation of eigencomponents. The method integrates fPCA, random measure theory, and nonparametric/semiparametric estimation, yielding closed-form solutions for Poisson and Hawkes processes. Evaluated on seismological, single-cell spatial transcriptomic, and neurophysiological datasets, our approach significantly improves both the accuracy of identifying population-level variation structures in point patterns and their biological interpretability.

This study investigates the finite-time queue peak dynamics in resource-constrained stochastic networks, focusing on generalized switch networks satisfying a uniform interior slack condition. Employing drift-minimizing scheduling policies such as MaxWeight, it reveals—for the first time—a two-phase growth pattern of queue peaks: sublinear square-root growth up to a geometrically determined threshold, followed by a universal logarithmic growth thereafter. The work introduces a novel self-normalization mechanism to explain this phenomenon and rigorously establishes the inevitability of both the logarithmic term and the geometric threshold. By integrating state-space collapse with local bottleneck geometry, the threshold can be optimized accordingly. Through stochastic process analysis and geometric methods rooted in capacity region theory, the paper derives tight upper bounds on peak queue lengths and precisely characterizes the logarithmic coefficient, with simulations confirming the two-phase envelope, geometric refinements, and heightened sensitivity to traffic variability.

This study addresses the challenge of quantifying optimal timing for proactive defense in single-attack scenarios by proposing a stochastic-process-based analytical framework, which introduces stochastic game theory into cybersecurity for the first time. By modeling defense as a continuous observation mechanism and integrating exponential distributions, Markovian Poisson arrival processes, Laplace–Carson transforms, and first-exit theory, the work explicitly derives the marginal distribution and conditional expectation of defensive actions. A joint detection function is constructed to precisely localize the attack instant. The approach not only enables visualization of defense density but also provides conditional expectations of observation times before and after an attack, facilitating dynamic calibration of low-latency proactive defense parameters according to threat intensity.

This study investigates the long-term evolutionary dynamics of heterogeneous economic agents characterized by asymmetric entry and exit mechanisms. By constructing a generalized geometric Brownian motion model incorporating asymmetric inflow and outflow rates, and employing stochastic process theory alongside non-conservative system analysis, the authors derive the stationary distribution and characterize the dynamics of moments as well as first-passage time properties. The primary contributions include uncovering three distinct regimes of moment evolution governed by the interplay among volatility, drift, and entry–exit rates, demonstrating that the system nonetheless converges to a well-defined stationary distribution, and identifying an optimal exit rate that significantly reduces the first-passage time. These findings quantitatively elucidate the critical influence of entry and exit strategies on systemic dynamics.

This work addresses the lack of efficient numerical methods for simulating Brownian motion and performing Langevin sampling on metric graphs. We propose the first time-step-split Euler–Maruyama discretization scheme, which analytically decouples intra-edge diffusion from vertex-jump dynamics to design a splitting strategy that preserves both convergence and jump-probability consistency—thereby overcoming the restrictive time-step constraints inherent in conventional finite-volume methods. Leveraging custom CUDA kernels, we achieve highly parallel GPU acceleration: up to 8,000× speedup on star graphs and 1,500× over DuMuX on realistic cortical vascular networks, while enabling significantly larger stable time steps. To our knowledge, this is the first scalable, high-accuracy, and computationally efficient numerical framework for Brownian motion simulation and sampling on metric graphs.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.