Koopman operator-based discussion on partial observation in stochastic systems

It is sometimes difficult to achieve a complete observation for a full set of observables, and partial observations are necessary. For deterministic systems, the Mori-Zwanzig formalism provides a theoretical framework for handling partial observations. Recently, data-driven algorithms based on the Koopman operator theory have made significant progress, and there is a discussion to connect the Mori-Zwanzig formalism with the Koopman operator theory. In this work, we discuss the effects of partial observation in stochastic systems using the Koopman operator theory. The discussion clarifies the importance of distinguishing the state space and the function space in stochastic systems. Even in stochastic systems, the delay-embedding technique is beneficial for partial observation, and several numerical experiments show a power-law behavior of error with respect to the amplitude of the additive noise. We also discuss the relation between the exponent of the power-law behavior and the effects of partial observation.