Long-time asymptotics for multivariate Hawkes processes with long-range interactions

This paper investigates the long-time asymptotic behavior of multivariate Hawkes processes with power-law decaying interactions, employing a combination of short-range interaction techniques, $\alpha$-stable law properties, and Tauberian theorems to model realistic systems like neural networks with long-range connections.

The Gist

Imagine a giant, invisible city where every building is a "particle" (like a neuron in a brain or a person in a social network). In this city, when one building "lights up" (an event happens), it sends a signal to its neighbors. Those neighbors might light up too, which sends signals to their neighbors, creating a chain reaction.

This paper studies a specific type of city called a Hawkes Process. The big question the author, Nadia Belmabrouk, is asking is: How does this city behave after a very, very long time?

Here is the breakdown of the paper using simple analogies:

1. The Two Types of Connections

In most previous studies, scientists assumed that buildings only talked to their immediate neighbors (the house next door). This is like a game of "telephone" where you only whisper to the person standing right next to you.

However, in the real world (especially in brains or social media), connections can be long-range. A building in New York might influence a building in London, even though they are far apart.

• The Twist: In this paper, the strength of these long-distance whispers gets weaker the further away they are, following a specific mathematical rule (a "power law"). It's like shouting across a canyon: the further you are, the quieter the voice, but it never completely disappears.

2. The Two Scenarios: Calm vs. Chaos

The paper looks at two different moods the city can be in, depending on how loud the "shouts" are:

Scenario A: The Calm City (Sub-critical Regime)

Imagine the whispers are quiet. When a building lights up, it might cause a few neighbors to light up, but the chain reaction eventually fizzles out.

• The Result: Over time, the city settles into a predictable rhythm. The average number of lights in any building stabilizes.
• The Math: The author proves that even with long-distance whispers, if the noise is low enough, the system behaves nicely and settles down to a steady state. It's like a crowd that eventually stops clapping and returns to silence.

Scenario B: The Wild Party (Super-critical Regime)

Now, imagine the whispers are very loud. When one building lights up, it triggers a massive explosion of activity that spreads faster than it dies out. The city goes into a frenzy.

• The Problem: In previous studies, scientists used a specific "shortcut" (a mathematical trick) to predict what happens in this wild scenario. But that shortcut breaks when you have long-distance connections. It's like trying to predict a hurricane by only looking at the wind speed in your backyard; you miss the bigger picture.
• The Solution: The author had to invent a new tool called a Tauberian Theorem.
  • Analogy: Imagine you are trying to guess the total weight of a giant, invisible elephant by looking at the shadow it casts. The old tools couldn't handle the elephant's long, weird shadow. The author used a new "shadow-measuring" technique (Tauberian theorems) to figure out the elephant's weight anyway.
• The Result: Even in the wild, chaotic regime, the city doesn't just explode into nonsense. It grows at a very specific, predictable speed (exponential growth). The author calculated exactly how fast it grows based on the strength of the long-distance connections.

3. Why Does This Matter?

You might ask, "Who cares about mathematical cities?"

This model is crucial for understanding Neural Networks (how our brains work).

• In a brain, neurons don't just talk to the ones touching them; they have long-range connections across the brain.
• If we want to understand how a brain processes information, or how a seizure (a massive chain reaction of neurons) starts and spreads, we need to understand these "long-range" interactions.
• This paper provides the mathematical rules to predict whether a brain will stay calm or spiral into a seizure, even when the connections are complex and far-reaching.

Summary

• The Old Way: Studied systems where things only affect their immediate neighbors.
• The New Way: Studies systems where things affect far-away neighbors (like a brain).
• The Discovery:
  1. If the system is quiet, it settles down nicely.
  2. If the system is loud, it grows explosively, but at a predictable rate.
  3. The author had to use advanced math (like a new type of telescope) to see the pattern in the loud scenario because the old tools didn't work.

In short, this paper gives us a better map for understanding how complex, interconnected systems (like our brains or social networks) behave over the long haul, whether they are calm or going wild.

Technical Summary

Here is a detailed technical summary of the paper "Long-time asymptotics for multivariate Hawkes processes with long-range interactions" by Nadia Belmabrouk.

1. Problem Statement

The paper investigates the long-time asymptotic behavior of multivariate Hawkes processes defined on an infinite lattice ($i \in \mathbb{Z}$) where the interaction between particles exhibits long-range dependence.

• Standard Context: Hawkes processes model self-exciting point processes where the occurrence of an event increases the probability of future events. Multivariate versions model interacting systems (e.g., neural networks, financial order books).
• The Gap: Previous work (specifically reference [3]) analyzed these systems only under short-range interactions (nearest-neighbor) or specific summable conditions.
• The Novelty: This paper introduces a model where the interaction strength decays as a power law with respect to the distance between particles $|i-j|$. The interaction kernel is defined as:
  $$\phi_{ji}(t) \propto \frac{\phi(t)}{|i-j|^{1+\alpha}}$$
  where $\alpha > 0$. This creates a regime where interactions are non-local, potentially leading to non-summable expectations or variances depending on $\alpha$.

The paper aims to characterize the asymptotic behavior of the expected number of events $E[Z^i_t]$ as $t \to \infty$ in both sub-critical and super-critical regimes.

2. Mathematical Model and Assumptions

The system is defined by the stochastic intensity $\lambda^i_t$ for particle $i$:
$$\lambda^i_t = \mu_i + \sum_{j \neq i} \frac{c(\alpha)}{|i-j|^{1+\alpha}} \int_0^t \phi(t-s) dZ^j_s$$
where:

• $\mu_i$ is the baseline intensity (bounded sequence).
• $\phi(t)$ is the excitation kernel (non-negative, measurable).
• $Z^i_t$ is the counting process for particle $i$.
• $A_\alpha$ is the interaction matrix with entries $A_\alpha(i,j) \propto |i-j|^{-(1+\alpha)}$.

Key Assumptions:

• Assumption 2.1: $\phi$ is measurable/non-negative; $\mu_i$ is bounded.
• Regime Definitions:
  • Sub-critical: $I = \int_0^\infty \phi(t) dt < 1$.
  • Super-critical: $I > 1$.
• Super-critical Constraint: To handle the growth, $\phi(t)$ is assumed to be sub-exponential ($\phi(t) \leq Ce^{\kappa t}$) rather than just sub-polynomial.

3. Methodology

The proofs combine techniques from probability theory, stochastic analysis, and asymptotic analysis:

1. $\alpha$-Stable Laws and Random Walks:

   • The interaction matrix $A_\alpha$ is treated as the transition kernel of a random walk on $\mathbb{Z}$.
   • The paper utilizes the Generalized Central Limit Theorem for random walks with step distributions in the domain of attraction of an $\alpha$-stable law.
   • It establishes that the $n$-th power of the matrix $A_\alpha^n$ converges to the density of an $\alpha$-stable distribution ($p_n$) as $n \to \infty$.

2. Laplace Transforms and Tauberian Theorems:

   • In the super-critical regime, the standard reduction to a 1D mean equation (used in previous literature) fails because the interaction is long-range.
   • The author employs Tauberian theorems (specifically relating the behavior of a function at infinity to the behavior of its Laplace transform near a singularity) to derive the asymptotic growth rate.
   • The analysis focuses on the singularity of the Laplace transform of the kernel $\hat{\phi}(p)$ at a specific point $\theta$ where $\hat{\phi}(\theta) = 1$.

3. Martingale Decomposition:

   • The error term between the process $Z^i_t$ and its expectation $m^i_t$ is analyzed using martingale techniques to prove convergence in mean.

4. Key Results

A. Sub-critical Regime ($I < 1$)

Theorem 2.2:
In the sub-critical regime, the process stabilizes. The expected number of events grows linearly with time, converging to a spatial average determined by the interaction matrix and baseline intensities.
$$\lim_{t \to \infty} E\left[ \left| \frac{Z^i_t}{t} - \sum_{j \in \mathbb{Z}} Q^I_\alpha(i, j) \mu_j \right| \right] = 0$$
where $Q^I_\alpha = \sum_{n \geq 0} I^n A_\alpha^n$.

• Significance: This extends previous nearest-neighbor results to long-range interactions, showing that the system remains stable and the spatial distribution of activity is determined by the convolution of the baseline intensity with the interaction matrix.

B. Super-critical Regime ($I > 1$)

Theorem 2.3:
In the super-critical regime, the system exhibits exponential growth. The paper generalizes previous results by relaxing the kernel assumptions to sub-exponential growth.
$$\lim_{t \to \infty} E\left[ \left| \frac{Z^i_t}{e^{\theta t}} - \frac{\bar{\mu}}{\theta^2 \bar{m}} \right| \right] = 0$$
where:

• $\theta > 0$ is the unique solution to the characteristic equation $\hat{\phi}(\theta) = 1$ (where $\hat{\phi}$ is the Laplace transform of $\phi$).

• $\bar{\mu}$ is the spatial average of the baseline intensities.

• $\bar{m} = \int_0^\infty t \phi(t) e^{-\theta t} dt$.

• Significance: This result proves that despite the long-range, non-summable nature of the interactions, the system's growth rate is governed by the "mean-field" parameter $\theta$ and the spatial average of the baseline. The spatial heterogeneity averages out in the long-time limit.

5. Technical Contributions

1. Adaptation of Lemma 27 (Lemma 4.4): The author proves that the infinite-dimensional system is well-posed under long-range interactions by establishing bounds on the weighted sums of the interaction matrix powers, utilizing the properties of $\alpha$-stable densities.
2. New Asymptotic Lemma (Lemma 3.1): The paper proves that the powers of the interaction matrix $A_\alpha^n$ converge to the spatial average $\bar{\mu}$ when applied to bounded sequences. This relies on the convergence of random walks to $\alpha$-stable laws, a significant departure from the Gaussian limits used in short-range models.
3. Tauberian Approach for Super-criticality: Instead of relying on the specific reduction techniques of [3] (which fail for long-range), the author constructs a rigorous proof using Laplace transforms and Tauberian theorems to handle the exponential growth and the specific singularity structure of the long-range kernel.
4. Generalization of Kernel Assumptions: The paper moves from requiring sub-polynomial kernels to sub-exponential kernels, broadening the applicability of the model to more realistic scenarios (e.g., neural networks with heavy-tailed or rapidly decaying but non-zero long-range connections).

6. Significance and Applications

• Neuroscience: The model is explicitly motivated by neural networks where neurons are connected over long distances. The results suggest that even with complex, long-range connectivity, the global firing rate of the network follows predictable asymptotic laws (linear in sub-critical, exponential in super-critical).
• Mathematical Physics: It provides a rigorous framework for interacting particle systems with power-law interactions, bridging the gap between mean-field theories and local interaction models.
• Methodological Advance: The successful application of Tauberian theorems and $\alpha$-stable limit theorems to multivariate Hawkes processes offers a new toolkit for analyzing systems with heavy-tailed dependencies, which are common in finance, seismology, and social dynamics.

In summary, Belmabrouk successfully extends the theory of Hawkes processes to the long-range regime, proving that the system's long-time behavior is robust and characterized by specific scaling laws derived from the interplay between the interaction decay exponent $\alpha$ and the excitation kernel $\phi$.