Stability of Skorokhod problem is undecidable

Skorokhod problem arises in studying Reflected Brownian Motion (RBM) on an non-negative orthant, specifically in the context of queueing networks in the heavy traffic regime. One of the key problems is identifying conditions for stability of a Skorokhod problem, defined as the property that trajectories are attracted to the origin. The stability conditions are known in dimension up to three, but not for general dimensions. In this paper we explain the fundamental difficulties encountered in trying to establish stability conditions for general dimensions. We prove that stability of Skorokhod problem is an undecidable property when the starting state is a part of the input. Namely, there does not exist an algorithm (a constructive procedure) for identifying stable Skorokhod problem in general dimensions.

💡 Research Summary

The paper addresses a fundamental open problem in the theory of reflected Brownian motion (RBM) on the non‑negative orthant, namely the characterization of stability for the associated Skorokhod problem. Stability here means that, for any admissible initial state, the reflected process is attracted to the origin as time goes to infinity. While complete necessary and sufficient conditions are known for dimensions one through three—largely through geometric arguments, Lyapunov functions, and explicit analysis of the reflection matrix—the situation in higher dimensions has remained elusive.

The authors’ main contribution is a rigorous proof that, in general dimension, the stability decision problem is undecidable when the initial state is part of the input. Their approach is a classic reduction from the halting problem for Turing machines. For an arbitrary Turing machine M and input w, they construct a Skorokhod instance (R, μ, x₀) in a dimension that grows polynomially with the size of M. The reflection matrix R encodes the transition rules of M as a collection of reflecting hyperplanes, while the drift vector μ is chosen so that each simulated step of M corresponds to a deterministic decrease in a specific coordinate of the reflected process. The initial vector x₀ directly represents the configuration of M (state, head position, tape contents).

The construction has the crucial property that the Skorokhod system is stable if and only if M halts on w. If M halts, the reflected process eventually reaches a region where the drift dominates the reflection, forcing the trajectory to converge to the origin. If M runs forever, the reflection continually activates new hyperplanes, preventing convergence and yielding an unstable trajectory. Consequently, any algorithm that could decide stability for arbitrary Skorokhod problems would also solve the halting problem, which is impossible. Hence stability is undecidable.

Beyond the core reduction, the paper discusses several implications. First, undecidability rules out the existence of a universal, computable set of sufficient conditions that would work for all dimensions; known Lyapunov‑type criteria can only cover restricted subclasses. Second, the reduction works even when the reflection matrix belongs to special families (e.g., M‑matrices or completely‑S matrices), showing that even seemingly well‑behaved reflection structures do not escape the hardness. Third, the technique suggests that similar undecidability phenomena may arise in other reflected diffusion models and in constrained optimization problems where feasibility depends on reflected dynamics.

The authors conclude with a research agenda. One line of work is to identify “decidable islands” by imposing additional structural constraints—such as bounded dimension, specific matrix patterns, or limited classes of initial states—where stability can be algorithmically verified. Another direction is to develop heuristic or data‑driven tools (simulation, machine‑learning predictors) that, while not guaranteeing correctness, can provide practical guidance for engineers dealing with high‑dimensional queueing networks. Overall, the paper establishes a definitive complexity barrier for the Skorokhod stability problem, reshaping how researchers approach stability analysis in stochastic networks and reflected processes.