Beyond Poisson: First-Passage Asymptotics of Renewal Shot Noise

Area of Science:

• Physics
• Biology
• Finance
• Stochastic Processes
• Non-Markovian Dynamics

Background:

• First-passage time (FPT) is crucial for stochastic signals in diverse fields.
• Existing analytical FPT results are limited to Poisson (Markovian) renewal shot noise.
• Non-Poisson arrival statistics are prevalent in biological and physical systems.

Purpose of the Study:

• To derive a universal asymptotic formula for the mean FPT to a threshold for renewal shot noise.
• To analyze the impact of arbitrary arrival statistics and temporal correlations on FPT.
• To establish a general framework for extreme events in non-Markovian systems.

Main Methods:

• Derivation of a universal asymptotic formula for mean FPT.
• Analysis of renewal shot noise with arbitrary arrival statistics and exponential marks.
• Development of a novel exact expression for noise moments.
• Numerical confirmation of theoretical results.

Main Results:

• A closed-form expression for mean FPT (⟨T_{b}⟩) is presented.
• Temporal correlations in arrivals modulate the Arrhenius law.
• Bursty arrivals accelerate threshold crossings via universal scaling corrections.
• Non-bursty arrivals exhibit Arrhenius-like behavior, linking burstiness to scaling.

Conclusions:

• The mean FPT provides a complete asymptotic characterization of threshold crossings.
• The derived formula offers a general framework for analyzing non-Markovian systems.
• The study bridges the gap between theoretical models and real-world non-Poisson processes.