NOTE-728: Brémaud & Massoulié — Stability of nonlinear Hawkes processes

Contribution

Hawkes and Oakes had proved existence, uniqueness and stability for the linear, excitatory Hawkes process, with arguments “where the linearity of φ is crucial”. This paper drops linearity and the sign restriction on the kernel. The rate becomes φ(∫h(t−s)N(ds)) with φ nonnegative and h of either sign, and the paper gives conditions under which a stationary version exists, is unique, and attracts versions started from a given past. Because “both types of arguments (linear or branching) are not available in the generality we have placed the problem”, the proofs use stochastic intensities, Poisson embedding (thinning) and coupling instead of the cluster representation.

Key insight

What makes a self-exciting process settle is not that it is linear or excitatory but that its rate is a contraction of its past. If φ moves by at most α per unit of input, and the kernel’s total absolute weight is h, then α∫|h| < 1 makes each pass of the past through the dynamics shrink differences, and a Picard-style iteration converges to the one stationary regime. In the linear excitatory case this is exactly the branching ratio being below one. And if φ is bounded, no such limit on the kernel is needed at all: a bounded rate cannot run away.

Assumptions

• Simple point processes on the whole real line, with an F_t-intensity λ(t) = φ(∫_(−∞,t) h(t−s) N(ds)) (eq. 1), φ: ℝ → ℝ⁺, h: ℝ⁺ → ℝ. The multivariate form (eq. 2) has one φ_i per component and a kernel h_ji from j to i, with no common points.
• Time-invariant kernels and rate functions. h and φ do not change with time or state beyond the dependence written in eq. 1. Nothing in the paper covers kernels that are themselves modulated.
• No marks, except a remark in the conclusion that independent marks are a straightforward extension (worked out in Massoulié’s thesis, not here).
• Stability is the paper’s own term (Definition 1): a stationary solution exists, and versions started from a past satisfying a stated initial condition converge to it in distribution or in variation. “Our use of the term stability in the title of this article does not seem to be standard in the theory of stochastic processes.”
• Uniqueness is among stationary solutions with finite average intensity (Theorems 1 and 7). Remark 6 notes that if φ(0) = 0 the empty process is a solution.
• All conditions are sufficient. No theorem gives a necessary condition, and none describes the process when the conditions fail.

Key results

Notation: α is a Lipschitz constant of φ, Λ a bound on the rate.

• Theorem 1 (unbounded Lipschitz dynamics). If φ is α-Lipschitz and α∫ℝ⁺ |h(t)| dt < 1: (a) there is a unique stationary distribution with finite average intensity; (b) the dynamics are stable in distribution from any past whose direct influence on the future vanishes (initial conditions (i) or (ii), in a.s. or in mean); (c) if also ∫ t|h(t)| dt < ∞, they are stable in variation from any past whose total direct influence ∫ dt ∫(−∞,0) |h(t−s)| N(ds) is a.s. finite (iii). Holds when: φ Lipschitz, h of either sign. The paper calls it “a strict generalization” of Hawkes and Oakes, whose linear case is φ(x) = ν + x, ν > 0, h ≥ 0, ∫h < 1, with stability proved only from an empty past.
• Remark 6. With φ(0) = 0 and ∫h < 1 in the linear case, there is no stationary process except the empty one, and any transient process “eventually dies out in distribution or in variation”. A self-exciting process below threshold needs an immigrant rate (ν > 0) to persist.
• Theorem 2 (bounded Lipschitz dynamics). If φ is α-Lipschitz and bounded by Λ, ∫|h| < ∞ and ∫ t|h(t)| dt < ∞, there is a unique stationary distribution, and the dynamics are stable in variation from any past whose influence on the far future tends to zero (eq. 5). There is no condition on the size of h. “at the expense of a boundedness condition” the L¹ limit is avoided. A steep ramp approximating a threshold is covered; the pure threshold is not: “Strictly speaking, however, this limiting case is not in the scope of Theorem 2.” The authors note that Kerstan (1964) had this result: “Our Theorem 2 is therefore of the same generality as the corresponding result of Kerstan, but does not contain more.”
• Theorem 3 (bounded memory dynamics). If the rate depends only on the last A time units of the past and is bounded by Λ, there is a unique stationary distribution, stability in variation holds “irrespective of the initial condition”, and convergence is exponentially fast. Compactly supported kernels give bounded memory. The proof uses the gaps longer than A in a dominating Poisson process of rate Λ as regeneration points.
• Theorem 4 (increasing kernel). If φ is nonnegative, nondecreasing, left-continuous and φ(x) ≤ λ + αx, and h ≥ 0 with α∫h < 1, a stationary process exists. No Lipschitz condition. Existence only: “it is not clear whether the stationary law for N is unique” (Remark 7). The construction gives the smallest stationary solution, and the largest when φ is continuous.
• Theorem 5. The multivariate form of Theorem 3: bounded rates with bounded memory give a unique stationary law, stable in variation from any past, exponentially fast.
• Theorem 6 (two neurons, monotone). For two components with bounded nondecreasing φ_i, nonnegative self-kernels, and cross-kernels that are both excitatory or both inhibitory, a stationary law exists. Existence only. Remark 9 extends this to K components split into two groups, excitatory within groups and inhibitory between them.
• Theorem 7 (multivariate Lipschitz). If each φ_i is α_i-Lipschitz and the K × K matrix A with entries a_ij = α_i ∫_0^∞ |h_ji(t)| dt has spectral radius strictly below one, there is a unique stationary law with finite average intensity, stable in distribution under (i′) or (ii′), and in variation under ∫ t|h_ji(t)| dt < ∞ and (iii′). The paper calls this “a strict extension of Hawkes and Oakes [7] results on linear mutually exciting point processes.”
• Theorem 8 (multivariate bounded Lipschitz). Bounded α_i-Lipschitz φ_i and integrable kernels with ∫ t|h_ij(t)| dt < ∞ give a unique stationary law, stable in variation from any past whose far-future influence vanishes (eq. 39). Again no condition on the size of the kernels.
• Examples. An M/D/K/0 loss queue (Example 1); threshold neurons, in which “Multivariate Hawkes processes also model neuronal activity and we shall therefore occasionally call them neural networks” (Example 2); and a network in which any recent spike silences a neuron (Example 3), which Theorem 7 covers because potentials are integer-valued (Example 5).

Claims

id	claim	strength	support
C1	Lipschitz φ, signed h, α∫|h| < 1 give a unique stationary law (finite mean intensity), stable in distribution, and in variation with a first-moment condition on h	strong	Theorem 1, proved
C2	Bounded Lipschitz φ gives a unique stationary law, stable in variation, with no limit on the size of h	strong	Theorem 2, proved (and Kerstan 1964)
C3	Bounded rate with bounded memory gives a unique stationary law and exponentially fast convergence from any past	strong	Theorem 3 and Theorem 5, proved
C4	Monotone φ with linear growth and h ≥ 0, α∫h < 1, gives a stationary law; uniqueness open	strong (existence only)	Theorem 4, proved; Remark 7
C5	For networks, a spectral radius below one of the matrix α_i∫|h_ji| suffices for a unique, stable stationary law	strong	Theorem 7, proved
C6	Mixed excitation and inhibition with monotone bounded φ admits a stationary law when cross-connections are sign-structured	strong (existence only)	Theorem 6, proved for two components; Remark 9 sketches K
C7	The linear-case proofs depend on linearity, and the branching argument does not reach the nonlinear signed case	moderate	stated in §2, not proved as an impossibility

Method

• Poisson embedding. A process with intensity λ(t) is built as the points of a homogeneous Poisson process on ℝ² lying under the curve t ↦ λ(t) (Lemma 3), or by thinning a rate-Λ Poisson process (Lemma 2). Lemma 4 runs the other way: every process with a predictable intensity can be embedded so.
• Iteration. Start from λ⁰ ≡ 0, feed each process into eq. 1 to get the next. Under the Lipschitz condition the expected differences shrink geometrically (Theorem 1, eq. 16; Theorem 7, eq. 35, with the matrix A), “analogous in some way to Picard’s existence proof for ordinary differential equations”. Under monotonicity the iterates increase (Theorem 4).
• Coupling. The stationary and the transient process are built from the same Poisson process, and convergence in variation is shown by their coinciding after an a.s. finite time (Lemma 5).

Concepts

• Nonlinear Hawkes process. A point process with intensity φ(∫h(t−s)N(ds)). Hawkes’s process is the case φ(x) = ν + x with h ≥ 0.
• Excitation function φ, transfer function h. The paper’s names, from the neural reading. h_ji is the transfer function from j to i.
• Initial condition. A property of the past N restricted to (−∞, 0), typically that its direct influence on the future vanishes or sums to a finite amount.
• Stability in distribution / in variation. Convergence of the shifted future to the stationary process, weakly or in total variation.
• Bounded memory. The rate depends on the past only through the last A time units.

Connections

It generalizes Hawkes (1971; LIT-1036) and Hawkes and Oakes (1974; LIT-1028), both unread here. By its account, Hawkes and Oakes proved existence by “constructing the corresponding stationary point process as a Poisson branching process or cluster Poisson process”, and uniqueness and stability from an empty past. Its Theorem 2 is Kerstan’s (1964) result with a shorter proof. Theorems 3 and 5 follow from Lindvall (1988) and the authors’ own earlier work on random memory. The constructions are those of Lewis and Shedler (1976) and Ogata (1981) for simulation.

Bearing on the record

On QUESTION-091: the formal half of the question has a precise form here, and the empirical half is untouched.

• The record’s question asks whether the allocation of influence moves cascades nearer to or further from criticality, “as a branching ratio would show”. For the linear excitatory case the branching ratio below one is Theorem 1’s condition with α = 1. For a network it is Theorem 7’s spectral radius. So the quantity the question names is the stability margin of a known theorem.
• For the exchange’s model, with signed kernels and a nonlinear rate, the same matrix with |h_ji| and the Lipschitz constants gives a sufficient condition. It is not a branching ratio. The process has no cluster representation in general (the paper says the branching argument is not available), and inhibition enters through |h|, so it counts toward instability in the bound, not against it. The spectral radius of this matrix being near one does not mean the process is near anything; the bound may be loose.
• Bounded rates have no threshold. Theorems 2, 3, 5 and 8 give stability for any kernel size once φ is bounded. If the processes in the record’s model have saturating influence, as any psychological rate plausibly does, then “distances from criticality” has no meaning supplied by these results. A question about distances from criticality needs an unbounded or effectively unbounded regime, and the record should say which it assumes.
• The kernels here are fixed. The question and A6’s hypothesis are about an influence canvas that changes the kernels. Nothing in this paper covers kernels that vary with time or with state. A slowly varying kernel could be treated as a sequence of fixed regimes, and the paper’s general initial conditions are meant for that: “such general conditions are needed to take into account the previous behavior of the system before a change in the parameters”. But a kernel that the process itself modulates is a different model, and its stability is not proved here.

On CLAIM-163: a Hawkes model supplies a coupling and a control parameter, not an order parameter, and not for valuations.

• CLAIM-163 asks for a coupling, an order parameter and transitions. In a multivariate Hawkes model the coupling is explicit: the kernels h_ji by which events of one component change another’s rate.
• The spectral radius of A is a function of the couplings and the rate functions. It is a control parameter, set before the dynamics run. An order parameter is a collective variable read off the state (the relative phase, in CLAIM-163’s own example). Of the paper’s quantities, the stationary average intensity is the nearest thing, and the paper only bounds it (Theorem 4’s proof). So the record’s wording in QUESTION-091, that “the branching ratio is the coupling and order parameter”, runs the two together.
• Transitions: the paper proves results only on the stable side. It says nothing about what happens at or past spectral radius one, and Theorem 2 shows that a bounded rate removes the transition altogether. So it does not supply CLAIM-163’s third part.
• The coupling is between streams of events, not between an agent’s valuations, and it is excitation, not phase-locking. So a Hawkes model could meet CLAIM-163’s demand for cascades of activity, as the QUESTION says, and does nothing for the synchronization reading the claim is about. It neither supports nor weakens CLAIM-163.

On CLAIM-286. The thesis needs models with signed, mixed interactions among many processes to be well posed. Theorems 6 and 7 show that some such models are: with Lipschitz or monotone bounded rates and the stated conditions, a stationary regime exists, and under Theorem 7 it is unique. That is a precondition for fitting the thesis’s model, not evidence for the thesis. Distinct “regimes” in that thesis’s sense (habit, rumination) would need multiple stationary laws or slow changes of parameters; under Theorems 1, 2, 7 and 8 the stationary law is unique, so under those conditions the model has one regime per parameter setting. Theorem 4 and 6 leave uniqueness open, and Remark 7’s smallest and largest solutions are where multiple regimes could live.

Limitations

• Sufficient conditions only. The Lipschitz-weighted |h| bound can be far from sharp when excitation and inhibition offset.
• No result on cluster or cascade sizes, the distribution of the number of events an event triggers, or the spectrum (Hawkes 1971’s subject).
• Kernels and rate functions are fixed; no time-varying or state-modulated kernels, and no learning.
• Uniqueness is open under monotone dynamics without Lipschitz conditions (Remark 7); the multivariate monotone case is “more delicate” and proved only for two components.
• Pure threshold rate functions are outside Theorem 2, inside Theorem 3 only with bounded memory.
• No statistical content: nothing on estimating kernels or testing which regime holds.

Open questions

• When excitation and inhibition coexist, what is the sharp condition for a unique stationary regime? A necessary condition would turn the bound into a threshold.
• What happens when the kernels are modulated by the process’s own history (an influence canvas)? A stability theorem for such a model, or a counterexample, would say whether QUESTION-091’s “distances from criticality” are well defined there.
• Under monotone dynamics, can the smallest and largest stationary solutions differ? If so, the model has more than one stationary regime for one parameter setting, which is what a regime-switching reading of CLAIM-286 would need.