The Memory Hidden in Response Fluctuations: Trajectory-Level Fluctuation-Response Theory and Inequalities for Non-Markovian Jump Dynamics

Key points

For non-Markovian jump processes whose event rates depend on past history, it exactly decomposed each event into a predictable tendency and an unpredictable martingale increment.
It proved that martingale increments form a complete orthogonal basis for the fluctuations of all trajectory observables, with each coefficient matching the history-dependent response strength.
It enabled testing whether proposed memory coordinates sufficiently explain the response, using a nonnegative response-heterogeneity gap that remains after averaging over histories.
In BK ion-channel data, memory ranked by dwell-time correlations and memory ranked by responses could have opposite orderings, showing that correlations alone cannot identify important memory.
The experimental application relies on observational comparisons rather than interventions and on thresholded, finite-length event sequences, so it may be affected by hidden variables and missed events; explosive dynamics are outside the theory's scope.

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