A tempered fractional Hawkes framework for finite-memory drought dynamics

Abstract.

Meteorological droughts emerge from nonlinear land--atmosphere feedbacks and circulation anomalies, whose persistence and recurrence are not well represented by conventional stochastic models with exponentially decaying memory. Here we introduce a Tempered Fractional Hawkes Process (TFHP) to describe drought onsets as a self-exciting point process with algebraically decaying but ultimately finite memory. In this formulation, the conditional intensity of dry-spell initiation obeys a tempered fractional differential equation in the Caputo sense, where the kernel ϕ(t) ∝ t^−αe^−θt combines long-range dependence with exponential tempering that enforces dynamic stability. The model parameters have clear physical meaning: µ (baseline exogenous activity), κ (self-excitation strength), α (fractional memory order), and θ⁻¹ (finite-memory horizon). Using 43 years of daily ERA5 precipitation over continental Chile, we estimate spatial fields of (µ, κ, α) and derive an effective memory timescale τ_m = 1/θ_eff. Results reveal geographically organised persistence regimes: subtropical arid and high-latitude subpolar regions exhibit slowly decaying memory and strong endogenous reinforcement, whereas mid-latitude zones display faster relaxation and weaker feedbacks. The TFHP thus offers a parsimonious and physically interpretable representation of finite climatic memory, bridging fractional calculus, point-process theory, and nonlinear geophysical dynamics. Beyond drought analysis, it provides a general framework to quantify persistence, clustering, and resilience in non-Markovian environmental systems.