---
slug: self_exciting_process
name: Self-Exciting Process
one_liner: "A stream of events in which each occurrence raises the momentary rate of further occurrences, so history feeds forward into future intensity — the history-dependent, clustering dual of the memoryless Poisson process along the arrival-dependence axis."
type: prime
schema_version: 2
status: density_v2_draft
categories: [dynamical, systemic, mathematical]
origin_domain: point_process_theory
alternate_origin_domains: [seismology, epidemiology, quantitative_finance, neuroscience]
aliases: [hawkes_process, self_excitation, contagious_arrivals, endogenous_clustering, history_dependent_point_process]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-01 dualization, sub-flavor 1a) — the structural dual of `poisson_process` along the arrival-dependence axis (memoryless/independent vs history-dependent/self-reinforcing). Discovered in the op-01 full sweep 2026-07-03. Verified absent via search_prime + slug check. STATUS: RECOMMENDED ACCEPT."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). Did NOT go through Phase-A triage. SI/SF/similarity grades are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT yet written."
- "Phase-C action: re-similarity-check vs poisson_process / feedback / contagion / criticality / preferential_attachment to confirm it does not collapse into a generic positive-feedback prime; at incorporation draw the dual edge to poisson_process (arrival-dependence axis), a specialization edge from feedback (positive feedback on an event rate), and sibling edges to contagion / cascade if present. one_liner must be synced to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 4
  structural_abstraction: 4
  transfer_evidence: 4
  reasoning: 'The rule "each event raises the conditional intensity of subsequent events, so occurrences beget occurrences" is a statement about a point process''s dependence structure with no commitment to a medium, giving structural abstraction a 4 (slightly below a pure relational 5 because the natural statement is intensity-theoretic — a conditional-intensity function — rather than a bare graph relation). It recurs demonstrably across substrates: earthquake aftershock sequences (the Hawkes/ETAS models), epidemic contagion and viral-content spread, neuronal spike bursts, order-book trade and default clustering in finance, and retaliatory cycles in conflict — a domain breadth of 4. Transfer is strong and well-documented because a single mathematical object (the Hawkes conditional intensity) is imported near-verbatim across seismology, finance, epidemiology, and neuroscience, so transfer evidence is 4 and the composite is 4: a robustly substrate-independent dependence structure whose home statement is intensity-theoretic rather than purely relational.'
similarity_to_nearest_existing_prime:
  slug: poisson_process
  similarity: 0.6
structural_framed:
  label: structural
  aggregate: 0.08
  criteria: {vocab_travels: 0.1, evaluative_weight: 0.0, institutional_origin: 0.0, human_practice_bound: 0.0, import_vs_recognize: 0.15}
  rationale: 'Self-excitation is a property of a point process — each event raises the conditional intensity of the next — carrying no normative or institutional content; every diagnostic reads at or near zero. It has a formal (point-process theory) origin and no evaluative weight: a self-exciting stream is neither good nor bad in itself, only relative to what is arriving. The small non-zeros reflect that "Hawkes process" and "self-exciting" are mild terms of art one carries in, but the underlying structure (occurrences raising the rate of further occurrences) is recognized wherever it appears — aftershocks, spike bursts, viral cascades — with no human in the loop.'
phase_b_passes_completed: []
---
# Self-Exciting Process

## Core Idea

A self-exciting process is a stream of events whose *momentary rate of arrival depends on its own history*: each occurrence temporarily raises the intensity at which further occurrences happen, so events beget events. It is one pole of a single structural axis — *whether the arrival rate depends on the process's own past* — whose opposite pole is `poisson_process`, the memoryless stream in which the rate is fixed and each event is independent of every other. Where the Poisson process forbids clustering beyond chance and has no memory of how long since the last event, the self-exciting process is defined by exactly the coupling the Poisson process rules out: the conditional intensity at any instant is not a constant but a function that jumps up on each event and decays back down between them, so a burst of activity makes more activity imminently more likely.

The load-bearing invariants are three. First, there is a *conditional intensity* — a well-defined instantaneous rate that is a function of the events observed so far, not a constant. Second, there is *positive history-dependence*: an event increments that intensity (the excitation), so the process feeds forward on itself rather than resetting. Third, there is a *decay*: the excitation from any given event fades over time, so the intensity relaxes toward a baseline unless fresh events sustain it. From these three follow the pattern's signature behaviors and its characteristic trade. The payoff of recognizing self-excitation is that it explains *clustering that a null model of independent arrivals cannot* — the observed events arrive in bursts separated by quiet, and the burstiness is endogenous, generated by the process itself rather than by an external driver. The governing cost or risk is *cascade and criticality*: because each event can trigger further events, the process sits on a branching threshold — if the expected number of direct offspring per event (the branching ratio) approaches one, excitation barely dies out and the stream becomes near-critical, prone to large, long, self-sustaining bursts; above one it explodes. The prime is the clean statement of that structure — a rate that its own events pump up and time decays down — abstracted from any particular kind of event.

Read this way, self-excitation is one pattern across many substrates. In seismology it is the *aftershock*: a mainshock raises the probability of further quakes nearby, which themselves have aftershocks (the Hawkes and ETAS models). In epidemiology and social media it is *contagion*: an infection or a share increases the rate of subsequent infections or shares before the effect decays. In neuroscience it is the *spike burst*: a neuron's firing transiently raises its own and its neighbors' firing probability. In finance it is *clustered volatility and default*: a trade or a failure raises the short-run intensity of further trades or failures. In each case the recognizable move is identical — an event raises the conditional rate of its own kind, that boost decays, and the interplay of excitation and decay determines whether the process merely clusters or runs away.

## Notes

**Provenance.** Produced by op-01 dualization (sub-flavor 1a) as the structural dual of `poisson_process` along the arrival-dependence axis: memoryless/independent/constant-rate arrivals versus history-dependent/self-reinforcing/clustered arrivals. Two independent routes in the op-01 full sweep (2026-07-03) proposed it, and the manifest flags it STRONG. Verified absent via search_prime + slug check.

**For owner evaluation (keep/cut).** The case FOR is strong. This is the canonical, named dual of an existing prime along a clean axis (dependence of the arrival rate on history), it is a single well-understood mathematical object (the Hawkes conditional intensity) imported near-verbatim across seismology, finance, epidemiology, and neuroscience — exactly the multi-substrate transfer that marks a genuine structural prime — and it fills a real gap: the corpus has the memoryless-arrivals pole but not the self-reinforcing-arrivals pole. The specific risk to weigh is *screen overlap with `feedback`*: self-excitation is, at bottom, positive feedback applied to an event rate, so the question is whether it is a distinct prime or merely feedback-instantiated-on-a-point-process. The counter-argument for distinctness is that self-excitation adds structure feedback alone does not name — a *stochastic point process* substrate, a *conditional intensity with excitation-plus-decay*, and a sharp *branching/criticality* threshold (branching ratio ≷ 1) — and that its transferable content (aftershock clustering, contagion, burstiness) travels as this specific object, not as generic feedback. A secondary check: confirm it is not better held as a specialization of `contagion`/`cascade` or collapsed into `criticality`; the branching-ratio behavior is a consequence of self-excitation, not identical to it. Theorem-vs-prime is not a concern here — this is a structure, not a result.

**At incorporation.** Re-run similarity against `poisson_process`, `feedback`, `contagion`, `cascade`, `criticality`, and `preferential_attachment` to confirm no collapse. Draw the dual edge to `poisson_process` (arrival-dependence axis), a specialization/instantiation edge from `feedback` (positive feedback on a conditional intensity), and sibling edges to `contagion`/`cascade` and to `criticality` (the branching-ratio threshold) if those primes are present. Write a v1 draft and sync the `one_liner` across v1 and v2 per CLAUDE.md before merge.
