Dynamical role of control-stabilized fixed points in noisy systems

Determine the dynamical role of control-stabilized fixed points in noisy stochastic systems—fixed points that arise in the Hamiltonian instanton dynamics at nonzero response/control field π, reflecting a balance between control and relaxation, and that do not exist in the deterministic dynamics. Clarify how these control-stabilized fixed points influence the system’s stability, trajectories, and irreversible behavior under noise-driven (endogenous control) dynamics.

Background

The paper establishes a mapping between noise and endogenous optimal control in stochastic systems, showing that the response/tilt field π in the Doi–Peliti formalism acts as a control variable in the semiclassical limit. Within this framework, the authors identify control-stabilized fixed points that do not exist in deterministic dynamics and instead reflect a balance between control and relaxation.

These non-classical fixed points can exhibit distinctive behavior, including cyclic motion with non-smooth feedback. Although the authors argue on topological grounds that such fixed points are generic and capture essential irreversible dynamics, they explicitly note that their dynamical role in noisy systems remains unclear, motivating a precise characterization of their impact on system behavior.