Infinite ergodic theory and functional statistics of non-confined Feller process

In the study of how things move and change over time, scientists often rely on a concept called ergodicity. Imagine watching a single drop of dye spread through a glass of water. If you wait long enough, that single drop will eventually visit every part of the glass, and the average behavior of that one drop will perfectly match the average behavior of a million drops released at once. This is the standard rule for many physical systems: time and space are interchangeable, and a long enough look at one thing tells you everything about the whole. However, nature has exceptions. In certain complex environments, particularly those where movement is influenced by the very position of the object itself, this rule breaks down. In these rare cases, a system can wander off toward infinity, never settling into a stable pattern, and the average behavior of a single path becomes a random, unpredictable thing that changes from one experiment to the next.

This is the territory of infinite ergodic theory, a field dedicated to understanding systems that refuse to settle down. A key model for this behavior is the Feller process, a mathematical description of a particle moving with a specific kind of random push and pull. When this process is confined, it stays within a bounded area and behaves predictably. But when it is unconfined, the particle drifts endlessly outward, and its probability of being found at any specific spot never adds up to a whole number. For decades, scientists have understood the timing of these wanderings, but the spatial structure of where the particle actually spends its time has remained a mystery. How does the particle's history look across different locations? Do the times it spends at one spot correlate with the times it spends at another?

In a recent study, physicist Vicenç Mendez from the Universitat Autònoma de Barcelona has mapped out this hidden spatial landscape. By analyzing the unconfined Feller process, Mendez discovered that the particle's history is not a chaotic jumble of independent events. Instead, the time the particle spends at any location is governed by a simple, elegant structure. The research shows that the amount of time a particle spends at a specific point is the product of two distinct factors: a fixed shape determined by the location itself, and a single, random clock that ticks for the entire system simultaneously. This means that if you were to watch the particle at two different distances from its starting point, the fluctuations in how long it stays there would not be independent. They would rise and fall in perfect unison, driven by a single global random amplitude.

To reach this conclusion, Mendez calculated the statistical correlations between the time spent at two different points in space. In most physical systems, the connection between two points weakens as the distance between them grows. If you watch a particle at one spot and then at another far away, their behaviors are usually unrelated. Mendez found the opposite to be true for this specific type of unconfined motion. As time goes on, the correlation between the time spent at any two locations becomes a constant, universal number. This number does not depend on how far apart the locations are, nor does it depend on which specific points are chosen. Whether the two observation points are close together or very far apart, their normalized relationship remains exactly the same. This reveals a rigid, global coherence across the entire space, where the entire system moves as a synchronized whole rather than a collection of independent parts.

The study also clarifies what happens to the particle's total time spent in different regions. For small, bounded areas near the origin, the time spent there grows slowly and follows a specific statistical pattern known as the Mittag-Leffler distribution. This distribution describes a type of randomness where the average behavior is not enough to predict the outcome; the result of one experiment will differ significantly from another, even if they are run under identical conditions. This confirms that the system exhibits "weak ergodicity breaking," where the standard rules of averaging fail. However, the research also draws a sharp line in the sand. When the scientists looked at observables that grow very large—specifically, functions that depend on the particle's position raised to a power—the rules changed again. These non-integrable quantities do not follow the same synchronized, Mittag-Leffler pattern. Instead, they exhibit a different kind of self-similar scaling, dominated by extreme excursions far from the origin. This distinction proves that the synchronized global behavior is a specific feature of how the particle occupies space, not a universal law for all types of measurements in this system.

The implications of this work extend beyond the specific math of the Feller process. It provides a unified framework for understanding how systems with multiplicative noise—where the randomness depends on the state of the system—behave when they are not confined. The finding that a single random amplitude can synchronize the entire spatial field suggests that in these infinite systems, the "clock" of the process is a global property. It is not a collection of local timers ticking at different rates, but a single, fluctuating rhythm that dictates the occupation density everywhere at once. This insight helps resolve long-standing questions about the spatial structure of infinite ergodic systems, showing that even in a system that never settles, there is a profound and rigid order to how it explores the infinite domain. The work demonstrates that while the particle may wander forever, its path is not a random scatter of independent steps, but a coherent, synchronized journey across the entire landscape.