Universal Linear Response of the Mean First-Passage Time

Abstract: First-passage processes are pervasive across numerous scientific fields, yet a general framework for understanding their response to external perturbations remains elusive. While the fluctuation-dissipation theorem offers a complete linear response theory for systems in steady-state, it fails to apply to transient first-passage processes. We address this challenge by focusing on rare, rather than weak, perturbations. Surprisingly, we discover that the linear response of the mean first-passage time (MFPT) to such perturbations is universal. It depends solely on the first two moments of the unperturbed first-passage time and the mean completion time following perturbation activation, without any assumptions about the underlying system's dynamics. To demonstrate the utility of our findings, we analyze the MFPT response of drift-diffusion processes in two scenarios: (i) stochastic resetting with information feedback, and (ii) an abrupt transition from a linear to a logarithmic potential. In both cases, our approach bypasses the need for explicit determination of the perturbed dynamics, unraveling a highly non-trivial response landscape with minimal effort. Finally, we show how our framework enables a new type of experiment-inferring molecular-level fluctuations from bulk measurements, a feat previously believed to be impossible. Overall, the newly discovered universality reported herein offers a powerful tool for predicting the impact of perturbations on kinetic processes, and for inferring underlying fluctuations from response measurements.