Source
Brémaud and Massoulié (1996), LIT-1035, read in NOTE-728: Theorems 1 and 2 (one process), 7 and 8 (networks), with Remarks 6 and 7. Active because these are proved theorems.
What was actually shown
The process has intensity λ_i(t) = φ_i(Σ_j ∫_(−∞,t) h_ji(t−s) N_j(ds)), with φ_i ≥ 0 and kernels h_ji of either sign, fixed in time.
What each theorem assumes, and gives.
| result | rate function φ | kernels h | size condition | gives |
|---|---|---|---|---|
| Theorem 1 | α-Lipschitz, unbounded allowed | signed | α∫|h| < 1 | unique stationary law (finite mean intensity); stable in distribution; in variation if also ∫t|h| < ∞ |
| Theorem 7 | α_i-Lipschitz | signed, K × K | spectral radius of A, a_ij = α_i∫|h_ji|, below 1 | the same, for networks |
| Theorem 2 | α-Lipschitz and bounded | signed, integrable, ∫t|h| < ∞ | none | unique stationary law; stable in variation |
| Theorem 8 | α_i-Lipschitz and bounded | signed, integrable, ∫t|h_ij| < ∞ | none | the same, for networks |
Stability is convergence of a process started from a given past to the stationary one. It holds from any past whose direct influence on the future vanishes, in a stated sense, not only from an empty past. The proofs use Poisson embedding, a Picard-style iteration that contracts under the size condition, and coupling.
The linear excitatory case. With φ(x) = ν + x, ν > 0, and h ≥ 0, Theorem 1’s condition is ∫h < 1, the branching ratio below one. This is the Hawkes–Oakes setting, and the paper calls Theorem 1 “a strict generalization” of their result. Remark 6 adds that with no immigrant rate (ν = 0) and ∫h < 1, the only stationary solution is the empty one.
Weaker results nearby. Under monotone dynamics without a Lipschitz condition (Theorem 4, excitatory only), and for sign-structured networks with bounded nondecreasing rates (Theorem 6, two components), a stationary law exists, and its uniqueness is left open (Remark 7).
What this does not say
- It is not a threshold. Every condition is sufficient. Nothing is proved at or past spectral radius one, and nothing says the process explodes or changes regime there. When excitation and inhibition offset, the |h| bound can be far from sharp.
- The spectral radius here is not a branching ratio outside the linear excitatory case. The paper says the branching argument is “not available in the generality we have placed the problem”, and inhibition enters the bound through |h|, adding to it. Reading the quantity as how far cascades travel imports the linear picture into a model it was not proved for.
- A bounded rate has no critical point. Theorems 2 and 8 hold for any kernel size. A model whose processes saturate is stable whatever the allocation of influence, as far as these theorems go.
- It says nothing about cascade sizes, the spectrum (Hawkes 1971, LIT-1036, unread), or estimation.
- Fixed kernels only. Kernels that change with time or with the process’s own state are not covered.
Bearing on the record
- QUESTION-091 (answers, in part). The question’s formal premise is that a branching ratio measures how far cascades propagate. This result says where that premise holds and where it stops. For linear excitatory processes the branching ratio is the stability margin. For the exchange’s signed, nonlinear model, a spectral radius below one is enough for a single stable regime, but its distance from one measures nothing proved, and with saturating rates there is no criticality to be near. The empirical half, whether influence allocation moves cascade sizes, is untouched; and the influence-dependent kernels the question is about are outside the theorem.
- CLAIM-163 (no relation). A multivariate Hawkes model has an explicit coupling, the kernels. The spectral radius is a control parameter built from them, not an order parameter, which is a collective variable of the state. The paper proves nothing about transitions. So it supplies one of CLAIM-163’s three parts outright, a candidate control parameter for a second, and nothing for the third, and it does so for cascades of events, not for an agent’s valuations. It neither supports nor weakens the objection, which is about valuations.
- CLAIM-286 (no relation). Theorems 6 and 7 show some signed, many-process models are well posed. That is a precondition for the thesis’s model, not evidence for it. Where these theorems give uniqueness, one parameter setting has one stationary regime, so regimes in that thesis’s sense would need parameter change or the non-unique cases Remark 7 leaves open.