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Transition pathways for a class of degenerate stochastic dynamical systems with Lévy noise

Ying Chao, Pingyuan Wei

TL;DR

This work derives the Onsager--Machlup action for a class of degenerate stochastic differential equations driven by both Brownian and Lévy noise, using a Girsanov-based path representation and a Hamilton--Pontryagin variational framework. The OM functional, which includes a Lévy-term contribution, acts as a Lagrangian whose minimization yields the most probable transition pathway (MPTP) under degeneracy, rather than a classical Euler--Lagrange solution. The authors establish a general OM theory for such systems, prove a variational structure via HP equations, and illustrate the approach with Langevin-type dynamics, including an analytical solver for a quadratic potential and global MPTP computation by optimizing endpoint velocities. This framework enables precise prediction of likely transition paths in systems with heavy-tailed noise and degenerate driving, with potential applications to molecular dynamics and climate-related metastable transitions. The results extend existing non-degenerate OM theory to degenerate settings and offer practical means to analyze transition phenomena under Lévy fluctuations.

Abstract

This work is devoted to deriving the Onsager--Machlup function for a class of degenerate stochastic dynamical systems with (non-Gaussian) Lévy 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.

Transition pathways for a class of degenerate stochastic dynamical systems with Lévy noise

TL;DR

This work derives the Onsager--Machlup action for a class of degenerate stochastic differential equations driven by both Brownian and Lévy noise, using a Girsanov-based path representation and a Hamilton--Pontryagin variational framework. The OM functional, which includes a Lévy-term contribution, acts as a Lagrangian whose minimization yields the most probable transition pathway (MPTP) under degeneracy, rather than a classical Euler--Lagrange solution. The authors establish a general OM theory for such systems, prove a variational structure via HP equations, and illustrate the approach with Langevin-type dynamics, including an analytical solver for a quadratic potential and global MPTP computation by optimizing endpoint velocities. This framework enables precise prediction of likely transition paths in systems with heavy-tailed noise and degenerate driving, with potential applications to molecular dynamics and climate-related metastable transitions. The results extend existing non-degenerate OM theory to degenerate settings and offer practical means to analyze transition phenomena under Lévy fluctuations.

Abstract

This work is devoted to deriving the Onsager--Machlup function for a class of degenerate stochastic dynamical systems with (non-Gaussian) Lévy 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.
Paper Structure (12 sections, 2 theorems, 56 equations, 2 figures)

This paper contains 12 sections, 2 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Degenerate stochastic differential equations with Lévy noise
  4. Onsager--Machlup function
  5. General theory
  6. Onsager--Machlup theory for the degenerate SDE with Lévy noise
  7. The most probable transition pathway between arbitrary states
  8. Proof of the Theorem \ref{['theorem 3.1']}
  9. Applications to Langevin systems
  10. (Parametric) Hamilton--Pontryagin equations v.s. (4th-order) Euler--Lagrange equations
  11. The MPTP based on an analytical solver
  12. Conclusion

Key Result

Theorem 3.1

(Onsager--Machlup theory) Consider a class of degenerate stochastic systems in the form of (Equation-2) with the jump measure satisfying $\int_{|\xi|<1}\xi \nu(d\xi)<\infty$ and the initial state $z_0=(x_0,y_0)\in\mathbb{R}^2$. Assume that $f\in C_b^2(\mathbb{R}^2,\mathbb{R})$ and $g\in C_b^1(\mathb with the Onsager--Machlup function (of course, up to an additive constant) where $\phi=(\phi_1,\ph

Figures (2)

  • Figure 1: (Color online) MPTPs from $X(0)=-1$ to $X(2)=1$ under different initial velocity $y_0$ and final velocity $y_T$: $T=2$, $\gamma=3$, $\alpha=\beta=\frac{1}{2}$ and thus $\Lambda_{\alpha,\beta}\approx0.3989$. Red solid line: the global MPTP with $y_0=5.8078$ and $y_T=0.1904$ for which the action functional reaches $I_{\text{min}}=-\frac{\gamma T}{2}=-3$.
  • Figure 2: (Color online) Let $T=2$, $\gamma=3$, $\mu=0.8$, $\alpha=\beta=\frac{1}{2}$ and thus $\Lambda_{\alpha,\beta}\approx0.3989$. (a) The patterns of sample paths and the MPTP. The trajectories of $X$ are shown as the top of figure and the trajectories of $Y$ are shown as the bottom of figure. (b) Around 15 simulations of system (\ref{['Langevin1']}) with initial value $X(0) = -1$ and final value $X(2)=1$ are shown. It is evident that simulations are more likely to accumulate around the MPTP (red line).

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5