Action functional as an early warning indicator in the space of probability measures via Schrödinger bridge.

Critical transitions and tipping phenomena between two meta-stable states in stochastic dynamical systems are a scientific issue. In this work, we expand the methodology of identifying the most probable transition pathway between two meta-stable states with Onsager-Machlup action functional, to investigate the evolutionary transition dynamics between two meta-stable invariant sets with Schrödinger bridge. In contrast to existing methodologies such as statistical analysis, bifurcation theory, information theory, statistical physics, topology, and graph theory for early warning indicators, we introduce a novel framework on Early Warning Signals (EWS) within the realm of probability measures that align with the entropy production rate. To validate our framework, we apply it to the Morris-Lecar model and investigate the transition dynamics between a meta-stable state and a stable invariant set (the limit cycle or homoclinic orbit) under various conditions. Additionally, we analyze real Alzheimer's data from the Alzheimer's Disease Neuroimaging Initiative database to explore EWS indicating the transition from healthy to pre-AD states. This framework not only expands the transition pathway to encompass measures between two specified densities on invariant sets, but also demonstrates the potential of our early warning indicators for complex diseases.