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

TL;DR

This paper derives the Onsager--Machlup function for degenerate stochastic systems with Lévy and Brownian noise, providing a variational framework to identify most probable transition pathways, with analytical and numerical analysis of a kinetic Langevin system.

Contribution

It introduces a method to derive the Onsager--Machlup function for degenerate systems with non-Gaussian Lévy noise, extending existing theories to more complex stochastic dynamics.

Findings

Derived the Onsager--Machlup function for systems with Lévy noise.

Applied the Hamilton--Pontryagin principle to degenerate stochastic systems.

Analyzed a kinetic Langevin system both analytically and numerically.

Abstract

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.