- The paper derives the Onsager-Machlup function as a Lagrangian for jump-diffusion processes, extending classical diffusion results with an additional term for Lévy noise.
- It applies the Girsanov transformation to express the Radon-Nikodym derivative in path integrals and validates the approach with numerical experiments on double-well and linear potential systems.
- The study provides a robust framework for modeling transitions in systems affected by both Gaussian and non-Gaussian noise, offering insights relevant to fields like climate science and genetics.
The Onsager-Machlup Function as Lagrangian for the Most Probable Path of a Jump-Diffusion Process
Introduction
The paper "The Onsager-Machlup Function as Lagrangian for the Most Probable Path of a Jump-diffusion Process" (1812.06409), authored by Ying Chao and Jinqiao Duan, presents an advancement in the stochastic analysis of jump-diffusion processes. The work derives the Onsager-Machlup function for a class of stochastic dynamical systems influenced by both Gaussian Brownian noise and non-Gaussian Lévy noise. The Onsager-Machlup function acts as a Lagrangian to determine the most probable paths connecting metastable states in such a system. This derivation is accomplished through the application of the Girsanov transformation to measures induced by jump-diffusion processes. The resulting Lagrangian not only aligns with previously established results for pure diffusion processes but also introduces an additional term reflecting the impact of Lévy noise.
Preliminaries and Framework
The authors provide comprehensive preliminary material on Lévy motions, a type of stochastic process characterized by stationary and independent increments, which includes a drift vector, a Brownian motion term, and a Poisson random measure representing jumps. The paper uses the Lévy-Itô decomposition to express these jump-diffusion processes mathematically. The framework set by the authors involves considering a class of one-dimensional nonlinear stochastic differential equations characterized by both Brownian and Lévy processes, aiming to model complex systems where such stochastic fluctuations prevail.
Derivation of the Onsager-Machlup Function
Central to the paper is the derivation of the Onsager-Machlup function for systems driven by jump-diffusion. The authors utilize the Girsanov transformation to handle measures between different induced processes and rewrite the Radon-Nikodym derivative in terms of path integrals. This leads to the derivation of the Onsager-Machlup function for the described class of jump-diffusion processes. The additional term due to Lévy noise distinguishes this work from existing literature on diffusion-only processes.
Most Probable Path and Numerical Experiments
The paper explores the determination of the most probable path by minimizing the Onsager-Machlup functional over suitable spaces, providing conditions under which a minimizer exists. This path corresponds to the solution of an Euler-Lagrange equation derived from the OM functional, subject to given boundary conditions. The authors conduct numerical experiments using concrete systems, such as a stochastic double-well system and a system with linear potential, to demonstrate the application of their theory. These examples highlight the influence of non-Gaussian noise on the most probable paths, showing significant divergence from paths induced by purely Gaussian noise.
Conclusion
The research extends the utility of the Onsager-Machlup approach to systems with non-Gaussian noise, providing a novel tool for understanding stochastic dynamics in biological and physical systems where Lévy noises are relevant. The implications of this work are substantial as it offers a method for identifying paths of highest likelihood in scenarios where classical Gaussian assumptions are inadequate. The extension of this model to higher dimensions is also discussed, indicating a wide applicability in complex systems analysis.
The authors crucially note that their method holds for systems with both Gaussian and non-Gaussian noise but is contingent on a non-vanishing diffusion component. This suggests potential avenues for future research in systems driven purely by jump noise, a challenging task given the limitations of Girsanov transformations in achieving uniqueness in distribution for such processes.
In summary, this paper contributes significantly to stochastic dynamics by providing an enhanced Lagrangian framework for analyzing transition paths in systems perturbed by complex noise processes, with potential applications in fields ranging from climate science to genetics.