Abstract
We have developed the theory of KM2O-Langevin equations for stationary and non-degenerate flow in an inner product space. As its generalization and refinement of the results in [{14}], [{15}], [{16}], we shall treat in this paper a general flow in an inner product space without both the stationarity property and the non-degeneracy property. At first, we shall derive the KM2O-Langevin equation describing the time evolution of the flow and prove the fluctuation-dissipation theorem which states that there exists a relation between the fluctuation part and the dissipation part of the above KM2O- Langevin equation. Next we shall prove the characterization theorem of stationarity property, the construction theorem of a flow with any given nonnegative definite matrix function as its two-point covariance matrix function and the extension theorem of a stationary flow without losing stationarity property.
