Probability Flow Approach to the Onsager--Machlup Functional for Jump-Diffusion Processes

TL;DR

This paper introduces a novel probabilistic flow method to derive the Onsager--Machlup functional for jump-diffusion processes, overcoming longstanding technical barriers and providing explicit formulas for finite and infinite jump activity cases.

Contribution

It presents the first rigorous derivation of the Onsager--Machlup functional for jump-diffusion processes using a probabilistic flow approach, including finite and infinite jump activity.

Findings

Closed-form Onsager--Machlup functional for jump-diffusions with finite jumps.

Inclusion of Lévy intensity at the origin in the functional.

Time-discrete Onsager--Machlup functional for infinite jump activity.

Abstract

The Onsager--Machlup action functional is an important concept in statistical mechanics and thermodynamics to describe the probability of fluctuations in nonequilibrium systems. It provides a powerful tool for analyzing and predicting the behavior of complex stochastic systems. For diffusion process, the path integral method and the Girsanov transformation are two main approaches to construct the Onsager--Machlup functional. However, it is a long-standing challenge to apply these two methods to the jump-diffusion process, because the complexity of jump noise presents intrinsic technical barriers to derive the Onsager--Machlup functional. In this work, we propose a new strategy to solve this problem by utilizing the equivalent probabilistic flow between the pure diffusion process and the jump-diffusion process. For the first time, we rigorously establish the closed-form expression of the Onsager--Machlup functional for jump-diffusion processes with finite jump activity, which include an important term of the L\'{e}vy intensity at the origin. The same probability flow approach is further applied to the L\'{e}vy process with infinite jump activity, and yields a time-discrete version of the Onsager--Machlup functional.