🔢 mathematics

Sharp Inverse Stability of Boundary Regulators in Two-Sided Reflected Delay Dynamics

This paper establishes a sharp, margin-adaptive stability theory for the inverse problem of recovering individual boundary regulators from two-sided reflected delay dynamics, proving optimal Lipschitz recovery, pathwise synchronization to a random equilibrium, and uniform Wong–Zakai convergence under strict delay dissipativity.

Original authors: Ahmed F. Abdel Jalil, Ayad R. Khudair

Published 2026-09-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ahmed F. Abdel Jalil, Ayad R. Khudair

Original paper licensed under CC BY 4.0 (https://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 particle moving along a narrow, straight track, trapped between two walls. If it tries to cross a wall, it bounces back. In the world of mathematics, this is a classic problem of "reflection." For decades, scientists have been able to predict exactly where the particle will be at any moment, given its starting point and the random jiggles of the environment. They can also calculate the total force the walls exert on the particle over time. However, there is a hidden complication. The total force is a single number that represents the combined push from the left wall and the right wall. If the left wall pushes hard and the right wall pushes just as hard in the opposite direction, these forces cancel each other out in the final calculation. The result is a net force of zero, even though both walls were working very hard.

This creates a difficult puzzle for researchers. If you only look at the final, net result, you cannot tell how much each individual wall contributed. It is like seeing a bank account balance that hasn't changed; you cannot know if someone deposited a thousand dollars and withdrew a thousand dollars, or if no one touched the account at all. For a long time, mathematicians knew this "inverse problem" was unstable. They knew that tiny errors in measuring the particle's path could lead to massive, unpredictable errors when trying to figure out the separate actions of the two walls. This made it nearly impossible to study the individual behavior of the boundaries in complex systems, such as those involving time delays or random noise.

A team of mathematicians from the University of Basrah has now solved this puzzle, but with a crucial condition. They discovered that while the problem is generally unstable, it becomes perfectly solvable if you look closely at the specific path the particle takes. They found that the difficulty of separating the two wall forces depends entirely on how often the particle bounces back and forth between the walls. If the particle zig-zags wildly, touching the left wall, then the right, then the left again, over and over, the math becomes messy and hard to untangle. But if the particle's path is relatively calm, or if the bounces are spaced out in a predictable way, the individual forces can be recovered with high precision.

The researchers developed a new method to measure this "complexity" of the path. They created a rule that counts the minimum number of time segments needed to describe the particle's journey without it hitting both walls within the same segment. This count acts as a measure of how hard the math is to do. Their key finding is that the error in calculating the separate wall forces grows in direct proportion to this count. In other words, if the path is twice as complex, the difficulty of separating the forces doubles. This relationship is "sharp," meaning it is the best possible outcome; no method could do better than this linear growth. This means that as long as the path isn't infinitely chaotic, the individual actions of the walls can be determined accurately, provided you have a good enough measurement of the particle's position.

The team then applied this new understanding to systems where the particle's movement depends not just on its current position, but on where it was a moment ago. This "delay" is common in real-world systems, from traffic flow to biological processes. They proved that even with these delays, the individual wall forces can be tracked continuously and accurately. They showed that over long periods, the system settles into a stable, predictable pattern, and the separate wall forces evolve in a way that is tightly linked to this stability. This allows scientists to observe the history of the wall interactions without losing the precision of the overall system's behavior.

To test their theory, the researchers ran computer simulations using a specific type of random motion. They smoothed out the random noise to make it easier to calculate, then compared the results to the true, rough random motion. The simulations confirmed their theory: the separate forces from the left and right walls converged to the correct values at the same speed as the overall system. Even when the walls had different physical properties, causing them to push with different strengths, the method worked perfectly. The numerical data showed that the errors in the separate wall forces decreased steadily as the simulation became more precise, matching the mathematical predictions exactly.

This work changes how we can study confined systems. Previously, scientists had to choose between knowing the total state of a system or knowing the details of its boundaries, but not both with high precision. Now, they can recover the individual actions of the boundaries from the total state, as long as they account for the complexity of the path. This opens the door to more accurate models of physical and biological systems where boundaries play a critical role, ensuring that the hidden contributions of each side are no longer lost in the noise of cancellation.

Technical Summary: Sharp Inverse Stability of Boundary Regulators in Two-Sided Reflected Delay Dynamics

Problem Statement
The paper addresses a fundamental inverse problem arising in the theory of two-sided reflected stochastic processes on a compact interval $[0, A]$ . While the forward Skorokhod map, which maps an unconstrained input path to a reflected path and a net constraining term, is classically known to be uniformly stable, the inverse problem is ill-posed in the uniform topology. Specifically, the reflected equation retains only the signed difference of two non-decreasing boundary regulators ( $\ell_0$ and $\ell_A$ ), defined as $q = \ell_0 - \ell_A$ . The individual cumulative actions $\ell_0$ and $\ell_A$ are uniquely identifiable via the Jordan decomposition of the signed measure $dq$, but recovering them from the aggregate term $q$ is not uniformly stable. Small perturbations in the reflected state and the net constraining term do not necessarily imply small perturbations in the individual regulators; they can remain separated by a constant order due to arbitrary oscillatory cancellation between the two boundaries.

Methodology
The authors develop a deterministic, trajectory-local stability theory to quantify the conditions under which the individual regulators can be stably recovered from aggregate data.

Intrinsic Complexity Definition: The core of the methodology is the introduction of a "margin-adaptive boundary-separation complexity," denoted $C_A(y; T, d)$ . This quantity is defined as the minimum number of intervals in a partition of the time horizon such that the oscillation of the reflected path $y$ on each interval is strictly less than $A - 2d$ , where $d$ is the uniform distance between the reference trajectory and a perturbed trajectory. This complexity acts as an intrinsic condition number for the inverse operation.
Inverse Stability Estimates: Using this complexity, the authors establish a Lipschitz inverse estimate. They prove that the error in recovering the individual regulators is bounded by the product of the complexity and the error in the signed constraining term.
Transfer Principle: An abstract transfer principle is derived, showing that if a family of trajectories has controlled intrinsic complexity, any forward approximation rate for the reflected state and signed action transfers to the individual regulators without loss of asymptotic order.
Stochastic Delay Dynamics: The theory is applied to reflected stochastic delay equations (RSDEs). The authors utilize a Lamperti transformation to convert multiplicative noise into additive noise while preserving distinct diffusion coefficients at the two boundaries. They construct continuous random dynamical systems (cocycles) for both the state and the additive regulator components.
Long-Time Dynamics: Under a strict delay dissipativity condition, the authors prove pathwise exponential synchronization, showing that the random pullback attractor collapses to a singleton random equilibrium. They further establish the upper semicontinuity of finite-horizon state-regulator observation graphs as the noise smoothing parameter vanishes.
Numerical Validation: A numerical experiment using matched-grid sensitivity diagnostics and Wong-Zakai approximations is conducted to verify that the componentwise regulators inherit the same Brownian-modulus convergence rates as the aggregate state, even when endpoint diffusion values differ.

Key Contributions and Results

Sharp Inverse Stability: The paper proves that the inverse map from the signed constraining term to the pair of boundary regulators is not uniformly continuous but is locally Lipschitz with a constant determined by the margin-adaptive complexity $C_A$ .
Optimality: A sawtooth construction demonstrates that the linear dependence on the intrinsic complexity is optimal in order; no sublinear modulus can control the recovery uniformly over all continuous trajectories.
Transfer Theorem: The authors provide a mechanism-independent transfer principle. For families with controlled intrinsic complexity (e.g., those with Hölder continuous paths), the convergence rate of the aggregate signed action is inherited by the separate regulators in both moments and almost surely.
Random Dynamical Systems: For reflected delay equations, the paper constructs a continuous skew-product random dynamical system on an augmented phase space (state + regulators). It proves the existence of a compact pullback attractor for the state component and establishes that, under strict dissipativity, the system synchronizes to a singleton random equilibrium.
Wong-Zakai Convergence: By combining the Lamperti transform with the inverse transfer theorem, the paper establishes uniform Wong-Zakai convergence for both endpoint regulators in moments and almost surely. Crucially, this result holds without assuming equality of the diffusion coefficients at the two boundaries.
Numerical Confirmation: The numerical results confirm that the componentwise error scales with the Brownian modulus (approximately $\delta^{1/2}\sqrt{\log(1/\delta)}$ ), validating the theoretical transfer principle in a setting with recurrent two-sided boundary activity and distinct diffusion values.

Significance and Claims
The paper claims to resolve the quantitative instability of separating two opposing cumulative boundary actions after they have been compressed into a single signed coordinate. The significance lies in:

Identifying the Obstruction: Explicitly characterizing the structural obstruction (oscillatory cancellation) that prevents uniform inverse stability and providing the precise trajectory-dependent condition (intrinsic complexity) required for recovery.
Bridging Forward and Inverse Problems: Demonstrating that forward approximation bounds for the reflected state can be rigorously transferred to the individual regulators, provided the trajectory complexity is controlled.
Long-Time Dynamics: Integrating the inverse stability theory into the long-time behavior of stochastic delay systems, showing that while the cumulative regulators are unbounded additive observables, their observation graphs are stable and converge to the synchronized equilibrium.
Generality: The results apply to reflected delay equations with multiplicative noise (via Lamperti transform) and distinct boundary diffusion coefficients, extending previous results that often assumed simpler noise structures or identical boundary behaviors.

The authors note that the "sharpness" of the result refers to the optimal linear order of the complexity dependence, not a specific numerical prefactor. They also acknowledge that extending this inverse theory to multidimensional polyhedral domains remains an open problem due to the increased complexity of facewise interactions and linear dependencies in reflection directions.

Technical Summary: Sharp Inverse Stability of Boundary Regulators in Two-Sided Reflected Delay Dynamics

More like this