Each event spikes the conditional intensity λ*(t) = μ + Σᵢ φ(t−tᵢ). Branching ratio n = ∫φ(s)ds < 1 ensures stationarity. At n→1, the process becomes critical — infinite aftershock cascades.
Parameters
Kernel: φ(t) = α·e^{−βt}
Branching ratio: n = α/β
n < 1: subcritical, stationary n → 1: critical, bursty n > 1: explosive (unstable)
Mean rate: λ̄ = μ/(1−n)
Applications:
• Earthquake aftershocks
• Financial order flow
• Viral social cascades
• Neuronal spike trains