Nonlinear Hawkes Process Analysis

• Nonlinear Hawkes processes are point processes where the intensity depends on a nonlinear function of past events, enabling self-excitation, inhibition, or saturation.
• They employ Palm-space operator formulations and coupling techniques to rigorously establish existence, uniqueness, and ergodic properties under minimal conditions.
• The framework delineates regimes—sublinear, linear, and superlinear—highlighting when stationary laws exist versus when explosive behavior ensues.

A nonlinear Hawkes process is a class of simple point processes in which the stochastic intensity at any time depends on a nonlinear functional of the past configuration of event times. Generalizing the classical linear Hawkes process, nonlinear Hawkes models are defined by a nonnegative memory kernel $h$ and a continuous, nonnegative, typically sublinear or Lipschitz-continuous activation function $\Phi$ (or "link function") that controls how past events influence the current rate. This framework enables rich self-exciting, inhibitory, or saturating feedback effects that go beyond the purely linear mutual excitation found in the original formulation.

1. Mathematical Structure and Palm-Space Formulation

The general form of a stationary version of a nonlinear Hawkes process $N$ on $\mathbb R$ is given by

$$\lambda(t) = \Phi\left(\sum_{T_i < t} h(t - T_i)\right),$$

where $\{T_i : i \in \mathbb{Z}\}$ are the points of the process, $h : [0,\infty) \to [0,\infty)$ is a continuous, non-increasing memory kernel with $\alpha = \int_0^\infty h(u)\,du < \infty$, and $\Phi : [0,\infty) \to [0,\infty)$ is a continuous activation map with $\Phi(0) > 0$ and at most linear growth $$\beta = \limsup_{x \to \infty} \frac{\Phi(x)}{x} < \infty$$.

A key analytic advance is the Palm-space operator formulation. By coding configurations with the (two-sided) sequence of interarrival distances $x = (\dots, x_{-1}, x_0, x_1, \dots)$ (where $x_k$ is the time between the $(k-1)$-th and $k$-th event in Palm perspective), the nonlinear Hawkes property induces a Markov evolution operator: given a current sequence and an independent $E \sim \mathrm{Exp}(1)$,

$$\int_{0}^{X_1} \Phi\left(h(s) + \sum_{k \geq 1} h\left(s + \sum_{i=1}^{k} x_i\right)\right) ds = E,$$

where the updated spacing $X_1$ is prepended to the left, shifting the sequence. This defines a Markov kernel $K$ on spacing sequences. Invariant probability measures of $(S, K)$ characterize the process's stationary Palm law.

2. Existence and Uniqueness of Stationary Solutions

General Existence Criterion: The process admits a stationary law under the minimal condition

$\alpha \cdot \beta < 1,$

where $\alpha = \int_0^\infty h(u)\,du$ and $\beta = \limsup_{x\to\infty} \frac{\Phi(x)}{x}$, as opposed to the stronger earlier requirement of global Lipschitz continuity of $\Phi$ with $\|\Phi\|_{\mathrm{Lip}}$ such that $\alpha \|\Phi\|_{\mathrm{Lip}} < 1$. This relaxes the condition found in classical works (Brémaud–Massoulié). The proof uses coupling with a linear Hawkes process, monotonicity arguments, and Palm space techniques.

Exponential Memory Case: If $h(s) = e^{-s/\alpha}$ with $\alpha > 0$, the nonlinear process reduces to a one-dimensional Harris-ergodic Markov chain $Z_{n+1} = 1 + e^{-X_{n+1} / \alpha} Z_n$ with $X_{n+1}$ defined implicitly as above. The process is positive recurrent if $\alpha \, \beta_e < 1$, where

$$\beta_e = \limsup_{u \to \infty} \int_{u - 1}^{u} \frac{\Phi(s)}{s} ds,$$

guaranteeing a unique stationary law. The Palm measure is directly realized as the invariant distribution of this chain.

3. Palm Measure, Stationarity, and Invariant Distributions

Stationarity of the process is tightly linked to the existence of invariant measures in the Palm-space Markovian formulation. The Mecke–Palm formula relates expectations under the stationary law $Q$ and under the Palm law $\widehat Q$: $$E_Q\left[F(N)\right] = \lambda E_{\widehat Q}\left[\int_0^{T_1} F(\theta_s N) ds\right],$$ where $T_1$ is the first point after $0$. Shift-invariance under $\widehat Q$ is preserved, and the original stationary law $Q$ can be reconstructed via a special-flow construction from $\widehat Q$.

The Markov kernel $K$ on spacing sequences, as generated by the Palm operator, fully characterizes the stationary Palm measure. Existence (or non-existence) of invariant laws for $K$ thus determines the ergodic character of the nonlinear Hawkes process.

4. Super-Linear Activation and Non-Existence of Stationary Regimes

For super-linear activation (e.g., $\Phi(x) = (\nu + \beta x)^\gamma$ with $\gamma > 1$), no nontrivial stationary process exists. Starting from the empty past, solutions to the associated SDE accumulate events at an explosive rate: $$T_{n+1} - T_n \asymp n^{-\gamma} E_{n+1}/\beta^\gamma,\quad (\text{in law for }\gamma \geq 2),$$ resulting in rescaled point processes converging in distribution to a Poisson process of rate $\beta^\gamma$. This phenomenon formalizes the nonexistence of any stationary regime in the super-linear activation case and quantifies the resulting blowup behavior.

5. Coupling, Monotonicity, and Novel Proof Techniques

The Palm-space approach enables coupling arguments with linear Hawkes processes whenever $\Phi(x) \leq \nu + \beta_0 x$ and $\alpha \beta_0 < 1$, which helps control pathwise excursions using monotonicity of $h$ and $\Phi$. Lyapunov drift conditions and Harris recurrence arguments establish ergodicity in the exponential memory case by identifying suitable test functions (e.g., $F(z) = \int_1^{z-1} \Phi(u)/u du$) and minorization sets. Explosion and scaling in the super-linear regime are revealed by analyzing the asymptotics of the induced Markov chains, establishing non-recurrence and local Poisson convergence of rescaled interarrival times.

A significant conceptual advance is the shift from a functional fixed-point or contraction-mapping approach toward a Markov-operator and coupling paradigm, particularly minimizing the need for global regularity of $\Phi$.

6. Implementation, Computational Aspects, and Extensions

Simulation schemes for nonlinear Hawkes processes in this framework generally rely on sequential generation of interarrival times via the implicit integral equation in the Palm operator or by solving the associated continuous-time SDE with random Poisson marks. For exponential memory kernels, the process reduces to a Markov chain, significantly simplifying both theoretical and computational analysis; the invariant law can be obtained via standard ergodic Markov chain methods.

In sublinear ($\Phi(x) = o(x)$) or polynomially growing kernel cases, similar operator methods apply, though care is required in regimes close to criticality or explosion. For general memory kernels satisfying $\int_0^\infty h(u)\,du < \infty$ and activating functions with at most linear growth at infinity, stationary processes exist and can be constructed via these methods; the recursions and operator equations become higher-dimensional if $h$ is not exponential.

The Palm-space and operator approach is robust to modifications, such as including negative (inhibitory) kernels, multi-dimensional extensions, or more general point process settings, provided the required sub-linearity and summability conditions are met.

7. Summary Table: Existence and Scaling Regimes

Activation $\Phi(x)$	Memory $h$	Stationary?	Scaling/Explosion
Sublinear, $\beta = 0$	$\int h < \infty$	Always	Yes, stationary
At most linear, $\alpha \beta < 1$	$\int h = \alpha < \infty$	Yes	Quantitative invariant law via coupling and Palm-space operator
Superlinear, $\gamma>1$	Any	No	Accumulation, $T_{n+1} - T_n \sim n^{-\gamma}$, Poisson limit for rescaled process

The above consolidates the mathematical regimes in terms of the activation growth, kernel summability, and the associated ergodic or explosive character of the process.

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The Palm space operator framework, together with coupling and monotonicity techniques, provides both a sharpened theoretical understanding of nonlinear Hawkes process regimes and a set of practical tools for establishing existence, uniqueness, and ergodicity under minimal conditions on the activation and memory kernel; it further quantifies the mechanism and scaling of explosive regimes when subcriticality fails (Robert et al., 2022).