🌀 nonlinear sciences

Quasi-stationary and quasi-ergodic distributions in the Pelikan random map

This paper introduces a concrete example of a substochastic Markov chain derived from an open Pelikan random map that exhibits a spectrum of infinitely many quasi-stationary distributions with distinct escape rates, while also establishing the existence of unique quasi-ergodic distributions for certain parameters and validating these findings through numerical simulations.

Original authors: Samuel Brevitt, Rainer Klages

Published 2026-07-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Samuel Brevitt, Rainer Klages

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where people are constantly moving, but there's a secret exit door that occasionally swallows a dancer whole. In the world of physics and mathematics, scientists study these "open" systems to understand how things behave when they aren't perfectly closed off. Usually, if you wait long enough, a system settles into a predictable rhythm, like a pendulum slowing down to a stop or a crowd finding a comfortable average density. This steady state is called an "invariant measure." But what happens when the exit door is always open, and people keep leaving? In these "open" systems, the usual rules break down. Instead of a single, stable rhythm, the system might have many different ways of behaving, depending entirely on how the dancers started out. This paper dives into that chaotic, open world, exploring a specific mathematical model called the "Pelikan map," which acts like a random, bouncy ball bouncing on a line with a hole in it. The big question is: if you start with different groups of people, do they all eventually fade away in the same way, or can they each find their own unique, temporary rhythm before vanishing?

The authors of this paper, Samuel Brevitt and Rainer Klages, have built a concrete example of a system that does something quite surprising: it supports an infinite number of different "quasi-stationary distributions" (QSDs). Think of a QSD as a temporary, stable pose that a group of dancers can hold for a while before the exit door finally claims them all. In most systems, there is only one way to hold this pose. However, in this specific model, the researchers found that for almost any setting of the system's knobs (parameters), there isn't just one way to dance; there is a whole spectrum of ways. Each of these different "poses" has its own unique speed at which the dancers leave the floor. If you start with a specific arrangement of dancers, the system will lock into one of these specific rhythms and maintain it, losing people at a specific, constant rate, until the very end.

The system they studied is a simplified version of a "random map," which is like a game where a ball moves on a number line. Sometimes the ball moves left, sometimes right, based on a coin flip. But there's a twist: if the ball hits zero, it gets "reset" to a random new spot further down the line, but with a chance that it might disappear entirely (the "hole"). The researchers used math to prove that this simple game allows for a continuous range of these temporary, stable states. They didn't just guess this; they calculated the exact mathematical formulas for these states and then ran computer simulations to watch them in action. The simulations confirmed that if you start the game with a specific distribution of balls, the system stays in that distribution, and the balls disappear at the exact rate predicted by the math.

Perhaps the most fascinating part of their discovery is that the "escape rate"—how fast the balls vanish—isn't fixed by the rules of the game alone. Instead, it depends on how you set up the game at the very beginning. You can have the same game with the same rules, but if you start with a different crowd arrangement, the balls will vanish at a completely different speed. The paper shows that this isn't a glitch or a rare accident; it's a fundamental feature of this type of open system. They also looked at "quasi-ergodic distributions" (QEDs), which are like the average history of a single ball's journey before it disappears. They found that these average histories exist and are unique, but only when the system is in a specific "recurrent" mode where the balls keep coming back to the start often enough. When the system is too "transient" (balls leave too quickly), these average histories don't settle into a stable shape at all.

The researchers also tested how sturdy these temporary rhythms are. They took one of these special "poses" and gave it a little nudge, like shaking the dance floor or adding a bit of random noise. They found that some of these rhythms are surprisingly stable; even with the noise, the system tends to stay in its original groove. However, if the noise is too wild or changes the shape of the crowd too drastically, the system might drift toward a different rhythm or fail to find a stable one at all. This suggests that while these multiple rhythms are real, they have their limits. The paper concludes that this behavior, while peculiar, is a genuine mathematical reality for this class of systems. It challenges the old idea that open systems always settle into a single, predictable way of dying out, showing instead that the starting conditions can dictate the entire story of how a system fades away. This finding is significant because it means that in real-world systems that lose energy or particles—like radioactive decay, populations in a shrinking habitat, or particles in a particle accelerator—the rate at which things disappear might be controllable simply by how you arrange them at the start, without changing the underlying laws of physics.

Technical Summary: Quasi-stationary and quasi-ergodic distributions in the Pelikan random map

Problem Statement
The paper addresses the spectral properties of discrete-time, countable, substochastic Markov chains, specifically focusing on the existence and uniqueness of Quasi-Stationary Distributions (QSDs) and Quasi-Ergodic Distributions (QEDs). While the theory for finite matrices is well-established (guaranteeing a unique QSD and QED under irreducibility and aperiodicity), the infinite-dimensional case is less understood. In countable systems, the transfer operator is not necessarily compact, and its spectrum may not be a discrete set of eigenvalues. Previous literature suggests that while a unique QSD may exist for specific initial conditions (e.g., fixed states), the general situation for arbitrary initial distributions is complex, potentially allowing for multiple QSDs with distinct escape rates. The authors seek to identify a concrete, non-trivial system that exhibits a continuous spectrum of QSDs and to analyze the stability and uniqueness of these distributions.

Methodology
The authors introduce a specific open variant of the Pelikan random dynamical system, modeled as a biased random walk on the half-line ( $k \in \mathbb{N} \cup \{0\}$ ) with stochastic resetting and an escape mechanism. The system is defined by a transfer operator $M$ with two parameters: a bias $p$ (probability of moving right) and a hole size $\varepsilon$ (probability of escape upon resetting at state 0).

The methodology employs a combination of analytical and numerical techniques:

Analytical Derivation: The authors utilize generating functions ( $A(z) = \sum \mu_k z^k$ ) to solve the eigenvalue equation $M\mu = \lambda\mu$ . They derive explicit recurrence relations for the components of the eigenvectors $\mu_k$ and analyze the singularities of the generating functions to determine the asymptotic "shape" (decay rate) of the distributions.
Vere-Jones Formalism: The system is analyzed using the formalism of $R$ -recurrence and $R$ -transience for countable absorbing Markov chains. This involves calculating the radius of convergence $R$ of the generating function for first passage times to determine the principal escape rate $\lambda_u = 1/R$ .
Positivity Analysis: A critical step involves determining the range of free parameters (specifically the initial mass $\mu_0$ or $\alpha$ ) for which the resulting eigenvector $\mu$ is non-negative everywhere, a necessary condition for a valid probability distribution. This is achieved by analyzing the roots of the denominator of the generating function and numerically verifying non-negativity up to a convergence criterion.
Numerical Simulation: The theoretical predictions are verified through simulations. The authors evolve perturbed initial distributions using the normalized transfer operator to test stability. They also simulate ensembles of trajectories to estimate QEDs by plotting time-averaged histograms of surviving trajectories.

Key Contributions and Results

Continuous Spectrum of QSDs: The primary result is the demonstration that for generic parameter values $(p, \varepsilon)$ , the system supports a continuous spectrum of infinitely many distinct QSDs. Unlike finite systems, there is not a single unique QSD; rather, there is a family of QSDs indexed by a free parameter $\alpha$ (related to the initial mass at state 0). Each QSD corresponds to a distinct eigenvalue $\lambda(\alpha)$ and thus a distinct exponential escape rate $\gamma_{esc} = -\ln \lambda$ .
Principal Escape Rate and Regimes: The authors identify a "principal" escape rate $\lambda_u = 1/R$ , which corresponds to the fastest possible escape rate. The system transitions between $R$ -positive (analogous to positive recurrence) and $R$ -transient regimes depending on $p$ and $\varepsilon$ . The principal QSD is unique within its specific asymptotic "shape" class.
Asymptotic Shape as a Conserved Quantity: The paper establishes that the asymptotic decay rate (shape) of a distribution is a conserved quantity under the system dynamics. The state space is partitioned into invariant sets based on this shape. If an initial distribution belongs to a shape class containing a QSD, the system converges to that QSD. If no QSD exists in that class (e.g., in certain $R$ -transient regions for fast-decaying initial conditions), the system exhibits non-uniform convergence where the escape rate adopts the principal rate while the tail retains the initial shape.
Stability to Perturbation: Numerical simulations indicate that non-principal QSDs are marginally stable to random noise. While they do not always converge to a single point in the $\ell_1$ norm under all noise types, they maintain their characteristic shape and escape rate, resisting drift toward the principal QSD unless the noise fundamentally alters the asymptotic shape (e.g., "thick" noise or sampling noise that truncates the tail).
Existence and Uniqueness of QEDs: The authors analytically construct QEDs (defined as the element-wise product of the QSD and the left eigenvector of $M$ ) for the $R$ -positive regime. They demonstrate that a normalizable QED exists and is unique only when the system is $R$ -positive. In the $R$ -transient regime, the constructed QED diverges, implying no normalizable QED exists. Simulations confirm that in the $R$ -transient regime, time-averaged distributions do not converge to a stationary shape but become increasingly flat.

Significance and Claims
The paper claims to provide the first explicit and concrete example in the literature of a non-trivial dynamical system naturally exhibiting a continuous spectrum of QSDs, all supported across the entire phase space. This challenges the intuition that QSDs are unique for irreducible systems.

The authors highlight that this multiplicity implies that the long-term escape rate of the system is not an intrinsic property of the parameters alone but can be controlled or prescribed by varying the initial distribution of conditions. Consequently, other dynamical quantities, such as the Lyapunov exponent or metric entropy, may also depend strongly on the initial arrangement. The work bridges concepts from absorbing Markov chains, stochastic resetting, and deterministic dynamical systems (specifically the Pelikan map), offering a new perspective on intermittency and open systems. The authors note that while the result is supported by existing theoretical frameworks (Vere-Jones formalism), the explicit demonstration of multiple QSDs in a simple random map has been met with surprise and skepticism by experts in the field, underscoring the novelty of the finding.

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