Latent Interacting Particle Systems

• Latent Interacting Particle Systems are a class of stochastic processes where particles evolve independently but interact through rare boundary events, facilitating the approximation of quasi-stationary distributions.
• They combine standard SDE simulation with a jump rule that reinserts boundary-hitting particles using a 'copycat' mechanism, preserving the overall particle count and system coherence.
• This methodology is pivotal for modeling complex dynamics in fields like population biology and statistical physics, offering a practical alternative to traditional spectral analysis.

Latent Interacting Particle Systems (LIPS) are a class of stochastic processes in which a finite or countable collection of “particles” (which may represent individual agents, components of a system, or biological entities) evolve in time according to prescribed stochastic dynamics, but with an essential coupling: the evolution of each particle is influenced, directly or indirectly, by the “latent” state of the entire system—particularly through mechanisms that only activate under special events, such as boundary hitting or absorption. This construction enables the empirical law of the system to converge, under appropriate scaling, to nontrivial statistical limits (e.g., quasi-stationary distributions, Yaglom limits), even in situations where explicit calculation is infeasible due to unbounded drift or complex domains. These systems are pivotal for approximating conditional long-time behaviors of absorbed Markov processes, and find applications in population dynamics, statistical physics, and beyond (Villemonais, 2010).

1. Construction and Dynamics

A canonical latent interacting particle system, as realized in (Villemonais, 2010), consists of $N$ independent diffusions in a domain $U \subset \mathbb{R}^d$ governed by the SDE: $dX_t = dB_t - q(X_t)\,dt,$ with $q : U \rightarrow \mathbb{R}^d$ typically representing the gradient of a confining potential $V$ (i.e., $q = \nabla V$). Unlike standard absorbed diffusions, when a particle hits the boundary $\partial U$, it does not get absorbed or killed; rather, it is instantaneously “jumped” back into the interior at a location drawn according to an interaction kernel that depends on the current configuration.

The canonical jump rule is: $$\mathcal{J}^{(m,N)}(x_1,\ldots, x_N) = \frac{1}{N-1} \sum_{j \ne i} \delta_{x_j}$$ for the particle $i$ hitting the boundary, meaning it is relocated to the position of a uniformly chosen other particle in the system (“copycat jump”). Such a rule enforces a minimal yet effective interaction, preserving the total number of particles in the interior.

More general jump measures—possessing a certain degree of “attraction” away from the boundary—are admissible, provided they satisfy integrability and nondegeneracy conditions (Hypothesis 1 in the original work). The key structural feature is that system–level interaction is only triggered by rare (boundary-hitting) events, making the dependency structure “latent” within the otherwise independent evolution.

This construction guarantees both the existence of the process without finite-time explosions and exponential ergodicity. The latter follows from coupling arguments that control the simultaneous approach of multiple particles to the boundary by embedding each $X^i$ in a reflected diffusion $Y^i$ with $0 \le Y^i_t \le \phi_D(X^i_t)$, where $\phi_D$ measures distance to $\partial U$.

2. Yaglom Limits and Quasi-Stationary Distributions

The central application of LIPS in (Villemonais, 2010) is the numerical approximation of quasi-stationary distributions (QSDs), or Yaglom limits, for absorbed diffusions in unbounded domains with (possibly unbounded) drift. Given a stochastic process $X^{\infty}$,

$$dX^{\infty}_t = dB_t - \nabla V(X^{\infty}_t)dt, \quad X^{\infty}_0 \in U_{\infty},$$

with absorption at the boundary, under mild assumptions on $V$ (see Hypothesis H3), there exists a QSD $\nu_{\infty}$: $$\nu_\infty = \lim_{t \to \infty} P_x(X^{\infty}_t \in \cdot \mid t < \tau_{\partial}),$$ which describes the long-time behavior of the system conditioned on survival.

The simulation strategy is to approximate $U_{\infty}$ by a sequence of increasing bounded domains $U_m \nearrow U_{\infty}$, running the interacting particle system of size $N$ on each $U_m$, and considering the empirical stationary measure $\mathfrak{E}^{(m, N)}$: $$\mathfrak{E}^{(m,N)} := \frac{1}{N} \sum_{i=1}^N \delta_{x^i}$$ where $M^{(m,N)}$ is the invariant law on $(U_m)^N$. The central result is two–level convergence: $$\lim_{m \to \infty}\lim_{N \to \infty} \mathfrak{E}^{(m,N)} = \nu_\infty$$ in the weak topology. First, as $N \to \infty$, the empirical distribution becomes tightly concentrated around the QSD $\nu_m$ for $U_m$. Second, as $m\to \infty$, the sequence $\nu_m$ converges to the true QSD of the process on $U_{\infty}$, including for unbounded drifts and pathological domains where spectral analysis is intractable.

3. Numerical Implementation and Approximation Algorithm

Numerical computation of QSDs via LIPS comprises three critical steps:

• SDE Simulation: Integrate $N$ independent particles via an SDE with drift on $U_m$ using standard numerical schemes (Euler–Maruyama or higher–order methods), detecting and handling boundary crossings.
• Jump Rule Execution: Upon a boundary hit, reinsert the affected particle by randomly copying the position of another particle (or more generally, sample according to a prescribed admissible jump measure).
• Empirical Stationarity: After running the process for long enough to reach stationarity (guaranteed to occur exponentially fast), form the histogram of particle positions. This histogram serves as the empirical stationary measure, providing a direct numerical estimate of the QSD.

It is essential to choose $N$ large and $m$ sufficiently high to reach the desired approximation quality, and verification of tightness and convergence is supported analytically (see Theorem 2.4 in (Villemonais, 2010)). This approach transforms deep spectral questions into practical simulation procedures.

4. Applications in Stochastic Population Dynamics and Related Areas

The methodology is particularly relevant in biological population modeling, where QSDs govern long-term evolutionary or ecological behavior conditional on persistence. Three prototypical examples are given:

• Wright–Fisher diffusion: Modeling gene frequency evolution with absorption at extinction. The QSD, known from exact solutions, is perfectly matched by empirical histograms with $N = 10^4$, cut-off $m=1000$.
• Logistic Feller diffusion: Incorporates competition and extinction; numerically, the QSD mass shifts according to model parameters, matching biological intuition on survival/extinction regimes.
• Stochastic Lotka–Volterra Systems: Predator–prey/competition interactions in the positive quadrant; when drift is unbounded and the domain $\mathbb{R}_+^2$ is unbounded, the LIPS method yields nontrivial QSDs capturing co-existence, inaccessible to direct analysis.

Similar methodology applies to epidemic modeling, reaction networks, and market contagion where persistent survival (or failure) events shape long-term system behavior.

5. Mathematical and Numerical Results

The effectiveness of the LIPS approach is substantiated by numerical illustrations and rigorous analytic control over convergence and tightness of the empirical measures:

• For the transformed Wright–Fisher diffusion, the stationary empirical density mirrors the true analytical QSD to within plotting resolution.
• In the logistic Feller case, systematic variation of reproduction or competition coefficients in the simulation moves the bulk of the empirical QSD as predicted by deterministic considerations.
• For two–species Lotka–Volterra, changing interaction coefficients alters the localization (density and contour) of the QSD, visually tracking coexistence domains in a way that pure extinction statistics would obscure.

These findings collectively confirm robust convergence and exportability of the methodology.

6. Tradeoffs, Limitations, and Extensibility

While simple to implement, three main tradeoffs arise:

• Computational Scaling: Large $N$ and substantial “burn-in” periods are required for accuracy, especially in high dimensions.
• Choice of Cutoff Domain $U_m$: Approximation quality of $\nu_\infty$ depends on sufficiently large $m$; inappropriate truncation may bias the empirical stationary distribution.
• Jump Mechanism: While uniform copycat jumps are straightforward, the theory accommodates a range of admissible kernels provided certain regularity criteria are met; choice may be tailored to enhance convergence or physical relevance.

The LIPS methodology is extendable to processes with more general SDEs (e.g. anisotropic diffusion or state-dependent noise), alternative boundary regimes, and multidimensional absorbed processes where analytical characterization of the QSD is intractable.

7. Theoretical Significance and Perspectives

The construction of LIPS for Yaglom limit approximation shifts the focus from spectral and functional analytic existence results to concrete algorithmic procedures, supplying:

• Ergodicity and tightness under minimal assumptions,
• Explicit formulas for limiting distributions in settings inaccessible to traditional methods,
• A tool for understanding latent preservation of mass and the detailed conditional structure shaped by rare “catastrophic” events (boundary hits),

The method offers a unifying framework for numerical computation of conditional distributions in complex stochastic dynamics, relevant across mathematics, computational biology, and applied probability (Villemonais, 2010).