arXiv subjects

Qiao Huang

Publications and source records attributed to Qiao Huang.

At least 19 recordsLinked to original sources

arXiv2026

Invariant connections in geometric mechanics: reduction, nonlocality, and curvature effects

We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler--Poincaré and Lie--Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan--Schouten connections, this leads to an integro-differential Euler--Poincaré equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.

math.DG↗

arXiv2026

Geometric Methods for Stochastic Dynamical Systems

Geometric methods are indispensable for analyzing, predicting, and mitigating the complex behaviors inherent in nonlinear systems. In this regime, the most probable transition path minimizes the Onsager-Machlup action functional, marking the likeliest route across an energy barrier. Lifting the analysis from individual sample paths to the infinite-dimensional space of probability densities, the book recasts these transitions as Schrödinger bridges - optimal paths between boundary distributions defined by minimizing relative entropy - and shows that the Onsager-Machlup path emerges as a special case when metastable states are idealized as Dirac masses. Generalizing further through α-divergences, which connect to entropies and the thermodynamic cost of nonequilibrium transitions, it introduces information geodesics as the resulting optimal density paths, offering a unified geometric account of how complex systems move between metastable regimes under uncertainty.

math.DS↗

arXiv2026

A study of path measures based on second-order Hamilton--Jacobi equations and their applications in stochastic thermodynamics

This paper provides a systematic investigation of the mathematical structure of path measures and their profound connections to stochastic differential equations (SDEs) through the framework of second-order Hamilton--Jacobi (HJ) equations. This approach establishes a unified methodology for analyzing large deviation principles (LDPs), entropy minimization, and entropy production in stochastic systems. Second-order HJ equations are shown to play a central role in bridging stochastic dynamics and measure theory while forming the foundation of stochastic geometric mechanics and their applications in stochastic thermodynamics. The large deviation rate function is rigorously derived from the probabilistic structure of path measures and proved to be equivalent to the Onsager--Machlup functional of stochastic gradient systems coupled with second-order HJ equations. We revisit entropy minimization problems, including finite time horizon problems and Schrödinger's problem, demonstrating the connections with stochastic geometric mechanics. Furthermore, we present a novel decomposition of entropy production for stochastic systems, revealing that thermodynamic irreversibility can be interpreted as the difference of the corresponding forward and backward second-order HJ equations. Together, this work establishes a comprehensive mathematical study of the relations between path measures and stochastic dynamical systems, and their diverse applications in stochastic thermodynamics and beyond.

math-ph↗

arXiv2026

Enhancing Requirements Traceability Link Recovery: A Novel Approach with T-SimCSE

Requirements traceability plays an important role in ensuring software quality and responding to changes in requirements. Requirements trace links (such as the links between requirements and other software artifacts) underpin the modeling and implementation of requirements traceability. With the rapid development of artificial intelligence, more and more pre-trained language models (PLMs) techniques are applied to the automatic recovery of requirements trace links. However, the requirements traceability links recovered by these approaches are not accurate enough, and many approaches require a large labeled dataset for training. Currently, there are very few labeled datasets available. To address these limitations, this paper proposes a novel requirements traceability link recovery approach called T-SimCSE, which is based on a PLM -- SimCSE. SimCSE has the advantages of not requiring labeled data, having broad applicability, and performing well. T-SimCSE firstly uses the SimCSE model to calculate the similarity between requirements and target artifacts, and employs a new metric (i.e. specificity) to reorder those target artifacts. Finally, the trace links are created between the requirement and the top-K target artifacts. We have evaluated T-SimCSE on ten public datasets by comparing them with other approaches. The results show that T-SimCSE achieves superior performance in terms of recall and Mean Average Precision (MAP).

cs.SE↗

arXiv2026

Stability analysis of a branching diffusion solver for semilinear heat equations

Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the integrability of random functionals of branching processes in order to ensure the non-explosion of solutions. In this paper, we study the stability of a functional branching representation of PDE solutions by deriving sufficient criteria for the integrability of the multiplicative weighted progeny of stochastic branching processes. We also prove the uniqueness of mild solutions under uniform integrability assumptions on random functionals.

math.PR↗

arXiv2025

Binary Galton-Watson trees with mutations

We consider a multitype Galton-Watson process that allows for the mutation and reversion of individual types in discrete and continuous time. In this setting, we explicitly compute the time evolution of quantities such as the mean and distributions of different types. This allows us in particular to estimate the proportions of different types in the long run, as well as the distribution of the first time of occurrence of a given type as the tree size or time increases. Our approach relies on the recursive computation of the joint distribution of types conditionally to the value of the total progeny. In comparison with the literature on related multitype models, we do not rely on approximations.

q-bio.PE↗

arXiv2025

On the random generation of Butcher trees

The main goal of this paper is to provide an algorithm for the random sampling of Butcher trees and the probabilistic numerical solution of ordinary differential equations (ODEs). This approach complements and simplifies a recent approach to the probabilistic representation of ODE solutions, by removing the need to generate random branching times. The random sampling of trees is compared to the finite order truncation of Butcher series in numerical experiments.

math.CA↗

arXiv2025

Probabilistic representation of ODE solutions with quantitative estimates

This paper considers the probabilistic representation of the solutions of ordinary differential equations (ODEs) by the generation of marked random trees in which marks can be interpreted as mutant types in population genetics models. We present sufficient conditions on equation coefficients that ensure the integrability and uniform integrability of the functionals of random trees used in this representation. Those conditions rely on the analysis of a marked branching process that controls the growth of random trees and provide implicit lower bounds on the explosion time of the underlying ODE, thus providing a connection between branching process explosion and the existence and uniqueness of ODE solutions.

math.PR↗

arXiv2025

The stochastic Hamilton-Jacobi-Bellman equation on Jacobi structures

Jacobi structures are known to generalize Poisson structures, encompassing symplectic, cosymplectic, and Lie-Poisson manifolds. Notably, other intriguing geometric structures -- such as contact and locally conformal symplectic manifolds -- also admit Jacobi structures but do not belong to the Poisson category. In this paper, we employ global stochastic analysis techniques, initially developed by Meyer and Schwartz, to rigorously introduce stochastic Hamiltonian systems on Jacobi manifolds. We then propose a stochastic Hamilton-Jacobi-Bellman (HJB) framework as an alternative perspective on the underlying dynamics. We emphasize that many of our results extend the work of Bismut [Bis80, Bis81] and Lázaro-Camí \& Ortega [LCO08, LCO09]. Furthermore, aspects of our geometric Hamilton-Jacobi theory in the stochastic setting draw inspiration from the deterministic contributions of Abraham \& Marsden [AM78], de León \& Sardón [dLS17], Esen et al. [EdLSZ21], and related literature.

math.DG↗

arXiv2024

Large Deviations for Stochastic Differential Equations Driven by Semimartingales

We prove a large deviation principle for stochastic differential equations driven by semimartingales, with additive controls. Conditions are given in terms of characteristics of driven semimartingales, so that if the noise-control pairs satisfy a large deviation principle with some good rate function, so do the solution processes. There is no joint exponential tightness assumption for noise-control-solution triplets and no uniform exponential tightness assumption for noise.

math.PR↗

arXiv2023

The Most Probable Transition Paths of Stochastic Dynamical Systems: A Sufficient and Necessary Characterization

The most probable transition paths of a stochastic dynamical system are the global minimizers of the Onsager-Machlup action functional and can be described by a necessary but not sufficient condition, the Euler-Lagrange equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devoted to showing a sufficient and necessary characterization for the most probable transition paths of stochastic dynamical systems with Brownian noise. We prove that, under appropriate conditions, the most probable transition paths are completely determined by a first-order ordinary differential equation. The equivalence is established by showing that the Onsager-Machlup action functional of the original system can be derived from the corresponding Markovian bridge process. For linear stochastic systems and the nonlinear Hongler's model, the first-order differential equations determining the most probable transition paths are shown analytically to imply the Euler-Lagrange equations of the Onsager-Machlup functional. For general nonlinear systems, the determining first-order differential equations can be approximated, in a short time or for the small noise case. Some numerical experiments are presented to illustrate our results.

math-ph↗

arXiv2023

Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable Lévy noises, whose stable indexes may be different. The homogenizing index $r_0$ of slow components has a relation with the stable index $α_1$ of the noise of fast components given by $0<r_0<2-2/{α_1}$. By first studying a nonlocal Poisson equation and then constructing suitable correctors, we obtain that the slow components weakly converge to a Lévy process as the scale parameter goes to zero.

math.DS↗

arXiv2023

Large Deviations for SDE driven by Heavy-tailed Lévy Processes

We obtain sample-path large deviations for a class of one-dimensional stochastic differential equations with bounded drifts and heavy-tailed Lévy processes. These heavy-tailed Lévy processes do not satisfy the exponential integrability condition, which is a common restriction on the Lévy processes in existing large deviations contents. We further prove that the solution processes satisfy a weak large deviation principle with a discrete rate function and logarithmic speed. We also show that they do not satisfy the full large deviation principle.

math.PR↗

arXiv2023

From Second-Order Differential Geometry to Stochastic Geometric Mechanics

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton-Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton-Jacobi-Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton's equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler-Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton's equations. A stochastic Noether's theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schrödinger's problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics.

math-ph↗

arXiv2023

Homogenization of Non-symmetric Jump Processes

We study the homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.

math.PR↗

arXiv2023

Stochastic Geometric Mechanics in Nonequilibrium Thermodynamics: Schrödinger meets Onsager

We are describing relations between Schrödinger's variational problem and Onsager's approach to nonequilibrium statistical mechanics. Although the second work on reciprocal relations and detailed balance has been published in the same year (1931) as the first one, the impact of Schrödinger's idea has not yet been considered in the classical context of Onsager.

cond-mat.stat-mech↗

arXiv2021

On the Stochastic Singular Cucker--Smale Model: Well-Posedness, Collision-Avoidance and Flocking

We study the Cucker--Smale (C-S) flocking systems involving both singularity and noise. We first show the local strong well-posedness for the stochastic singular C-S systems before the first collision time, which is a well defined stopping time. Then, for communication with higher order singularity at origin (corresponding to $α\ge1$ in the case of $ψ(r)=r^{-α}$), we establish the global well-posedness by showing the collision-avoidance in finite time, provided that there is no initial collisions and the initial velocities have finite moment of any positive order. Finally, we study the large time behavior of the solution when $ψ$ is of zero lower bound, and provide the emergence of conditional flocking or unconditional flocking in the mean sense, for constant and square integrable intensity respectively.

math.DS↗

arXiv2021

Homogenization of Nonlocal Partial Differential Equations Related to Stochastic Differential Equations with Lévy Noise

We study the "periodic homogenization" for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic $α$-stable Lévy noise ($1<α<2$) which is nonlinear in the noise component. Our homogenization method is probabilistic. It turns out that, under suitable regularity assumptions, the limit of the solutions satisfies a nonlocal partial differential equation with constant coefficients, which are associated to a symmetric $α$-stable Lévy process.

math.AP↗