arXiv subjects

Pingyuan Wei

Publications and source records attributed to Pingyuan Wei.

13 recordsLinked to original sources

arXiv2026

Metastable Transitions in Dynamical Systems with both Time-varying Perturbations and Degenerate Noise

This paper investigates the persistence of maximum likelihood paths in degenerate stochastic differential systems and quantifies how small periodic perturbations modulate the metastable transition rate. Within the Freidlin--Wentzell large deviation framework, we reformulate the variational problem for MLPs as a Hamiltonian system via a partial Legendre transform. Under hyperbolicity and transversality conditions, we prove, using a geometric Melnikov method adapted to general time-dependent perturbations, that the corresponding heteroclinic connections persist for sufficiently small perturbations. For the periodic case, we derive a closed-form explicit expression for the rate change to first order in the forcing amplitude. Two illustrative examples are presented.

math.DS↗

arXiv2026

Onsager--Machlup functionals for McKean--Vlasov SDEs via Euler-type approximation

The Onsager--Machlup action functional provides a variational framework for characterizing the most probable transition paths of stochastic systems and plays an important role in the study of nonequilibrium fluctuations. Its extension to McKean--Vlasov stochastic differential equations is complicated by the intrinsic distribution dependence of the coefficients. In this paper, we address this difficulty by introducing an Euler-type approximation scheme based on classical, distribution-free stochastic differential equations. Combining the classical Onsager--Machlup theory with a convergence argument for the approximation sequence, we derive an explicit expression for the Onsager--Machlup functional associated with the McKean--Vlasov SDE. The proposed approach is constructive and extends to a broad class of distribution-dependent stochastic systems.

math.DS↗

arXiv2025

A parameterization method for quasi-periodic systems with noise: computation of random invariant tori

This work is devoted to studying normally hyperbolic invariant manifolds (NHIMs) for a class of quasi-periodically forced systems subject to additional stochastic noise. These systems can be understood as skew-product systems. The existence of NHIMs is established by developing a parameterization method in random settings and applying the Implicit Function Theorem in appropriate Banach spaces. Based on this, we propose a numerical algorithm to compute the statistics of NHIMs and Lyapunov exponents.

math.DS↗

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 L\'azaro-Cam\'i \& 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 Le\'on \& Sard\'on [dLS17], Esen et al. [EdLSZ21], and related literature.

math.DG↗

arXiv2025

Transition pathways for a class of degenerate stochastic dynamical systems with L\'evy noise

This work is devoted to deriving the Onsager--Machlup function for a class of degenerate stochastic dynamical systems with (non-Gaussian) L\'{e}vy noise as well as Brownian noise. This is obtained based on the Girsanov transformation and then by a path representation. Moreover, this Onsager--Machlup function may be regarded as a Lagrangian giving the most probable transition pathways. The Hamilton--Pontryagin principle is essential to handle such a variational problem in degenerate case. Finally, a kinetic Langevin system in which noise is degenerate is specifically investigated analytically and numerically.

math.DS↗

arXiv2023

Well-posedness and averaging principle for Lévy-type McKean-Vlasov stochastic differential equations under local Lipschitz conditions

In this paper, we investigate a class of McKean-Vlasov stochastic differential equations under Lévy-type perturbations. We first establish the existence and uniqueness theorem for solutions of the McKean-Vlasov stochastic differential equations by utilizing the Euler-like approximation. Then under some suitable conditions, we show that the solutions of McKean-Vlasov stochastic differential equations can be approximated by the solutions of the associated averaged McKean-Vlasov stochastic differential equations in the sense of mean square convergence. In contrast to the existing work, a novel feature is the use of a much weaker condition -- local Lipschitzian in the state variables, allowing for possibly super-linearly growing drift, but linearly growing diffusion and jump coefficients. Therefore, our results are suitable for a wider class of McKean-Vlasov stochastic differential equations.

math.PR↗

arXiv2021

Lyapunov Exponents for Hamiltonian Systems under Small Lévy Perturbations

This work is to investigate the (top) Lyapunov exponent for a class of Hamiltonian systems under small non-Gaussian Lévy noise. In a suitable moving frame, the linearisation of such a system can be regarded as a small perturbation of a nilpotent linear system. The Lyapunov exponent is then estimated by taking a Pinsky-Wihstutz transformation and applying the Khas'minskii formula, under appropriate assumptions on smoothness, ergodicity and integrability. Finally, two examples are present to illustrate our results. The results characterize the growth or decay rates of a class of dynamical systems under the interaction between Hamiltonian structures and non-Gaussian uncertainties.

math.DS↗

arXiv2021

Formulation of Stochastic Contact Hamiltonian Systems

In this work we devise a stochastic version of contact Hamiltonian systems, and show that the phase flows of these systems preserve contact structures. Moreover, we provide a sufficient condition under which these stochastic contact Hamiltonian systems are completely integrable. This establishes an appropriate framework for investigating stochastic contact Hamiltonian systems.

math.DS↗

arXiv2020

The role of slow manifolds in parameter estimation for a multiscale stochastic system with $α$-stable Lévy noise

This work is about parameter estimation for a fast-slow stochastic system with non-Gaussian $α$-stable Lévy noise. When the observations are only available for slow components, a system parameter is estimated and the accuracy for this estimation is quantified by $p$-moment with $p\in(1, α)$, with the help of a reduced system through random slow manifold approximation. This method provides an advantage in computational complexity and cost, due to the dimension reduction in stochastic systems. To numerically illustrate this method, and to corroborate that the parameter estimator based on the reduced slow system is a good approximation for the true parameter value of the original system, a prototypical example is present.

math.DS↗

arXiv2020

Maximal Likely Phase Lines for a Reduced Ice Growth Model

We study the impact of Brownian noise on transitions between metastable equilibrium states in a stochastic ice sheet model. Two methods to accomplish different objectives are employed. The maximal likely trajectory by maximizing the probability density function and numerically solving the Fokker-Planck equation shows how the system will evolve over time. We have especially studied the maximal likely trajectories starting near the ice-free metastable state, and examined whether they evolve to or near the ice-covered metastable state for certain parameters, in order to gain insights into how the ice sheet formed. Furthermore, for the transition from ice-covered metastable state to the ice-free metastable state, we study the most probable path for various noise parameters via the Onsager-Machlup least action principle. This enables us to predict and visualize the melting process of the ice sheet if such a rare event ever does take place.

math.DS↗

arXiv2019

Hamiltonian Systems with Lévy Noise: Symplecticity, Hamilton's Principle and Averaging Principle

This work focuses on topics related to Hamiltonian stochastic differential equations with Lévy noise. We first show that the phase flow of the stochastic system preserves symplectic structure, and propose a stochastic version of Hamilton's principle by the corresponding formulation of the stochastic action integral and the Euler-Lagrange equation. Based on these properties, we further investigate the effective behaviour of a small transversal perturbation to a completely integrable stochastic Hamiltonian system with Lévy noise. We establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero. Furthermore, we obtain the estimation for the rate of this convergence. Finally, we present an example to illustrate these results.

math.DS↗

arXiv2019

Invariant foliations for stochastic dynamical systems with multiplicative stable Levy noise

This work deals with the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Levy noise. We first establish the existence of stable and unstable foliations for this system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Finally, we illustrate our results in an example.

math.DS↗

arXiv2019

Transitions between Metastable States in a Simplified Model for the Thermohaline Circulation under Random Fluctuations

In this work, we study the impact of non-Gaussian alpha-stable Levy motion on transitions between metastable equilibrium states (or attractors) in a stochastic Stommel two-box model for thermohaline circulation (THC). By maximizing the probability density of the solution process associated with a nonlocal Fokker-Planck equation, we compute maximal likely pathways and identify corresponding maximal likely stable equilibrium states. Our numerical results indicate weakened THC may be induced by perturbation with very small noise intensity in a certain range of stability index. Moreover, larger noise intensity and larger stability index induce weakened THC within shorter bifurcation time.

physics.ao-ph↗