Transition pathways for a class of high dimensional stochastic dynamical systems with L\'{e}vy noise

TL;DR

This paper derives the Onsager-Machlup action functional for high-dimensional stochastic systems driven by Lévy noise and Brownian motion, enabling analysis of most probable transition pathways through variational methods.

Contribution

It introduces a novel derivation of the Onsager-Machlup functional for high-dimensional Lévy-driven systems using Girsanov transformation and Poincaré lemma, facilitating pathway analysis.

Findings

Derived Onsager-Machlup functional for Lévy and Brownian systems

Provided conditions for path representation in high dimensions

Analyzed transition pathways analytically and numerically

Abstract

This work is devoted to deriving the Onsager-Machlup action functional for a class of stochastic differential equations with (non-Gaussian) L\'{e}vy process as well as Brownian motion in high dimensions. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. The Poincar\'{e} lemma is essential to handle such path representation problem in high dimensions. We provide a sufficient condition on the vector field such that this path representation holds in high dimensions. Moreover, this Onsager-Machlup action functional may be considered as the integral of a Lagrangian. Finally, by a variational principle, we investigate the most probable transition pathways analytically and numerically.