On the theory of KM2O-Langevin equations for non-stationary and degenerate flows

Abstract

We have developed the theory of $K{M}_{2}O$-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 $K{M}_{2}O$-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 $K{M}_{2}O\text{-}$ 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.