PROBABILITY FLOW APPROACH TO THE ONSAGER-MACHLUP FUNCTIONAL FOR JUMP-DIFFUSION PROCESSES

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 the 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 deriving 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 includes an important term of the L\'evy intensity at the origin. The same probability flow approach is further applied to the L\'evy process with infinite jump activity and yields a time-discrete version of the Onsager-Machlup functional. Copyright © 2025 Society for Industrial and Applied Mathematics.