## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 55, Number 2 (2003), 523-563.

### On the theory of $K{M}_{2}O$-Langevin equations for non-stationary and degenerate flows

Masaya MATSUURA and Yasunori OKABE

#### 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.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 55, Number 2 (2003), 523-563.

**Dates**

First available in Project Euclid: 3 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1191419129

**Digital Object Identifier**

doi:10.2969/jmsj/1191419129

**Mathematical Reviews number (MathSciNet)**

MR1961299

**Zentralblatt MATH identifier**

1162.82312

**Subjects**

Primary: 60G25: Prediction theory [See also 62M20]

Secondary: 60G12: General second-order processes 82C05: Classical dynamic and nonequilibrium statistical mechanics (general)

**Keywords**

flow $\mathrm{K}\mathrm{M}_{2}\mathrm{O}$-Langevin equation non-stationarity property degeneracy property fluctuation-dissipation theorem

#### Citation

MATSUURA, Masaya; OKABE, Yasunori. On the theory of $\mathrm{KM}_{2O}$ -Langevin equations for non-stationary and degenerate flows. J. Math. Soc. Japan 55 (2003), no. 2, 523--563. doi:10.2969/jmsj/1191419129. https://projecteuclid.org/euclid.jmsj/1191419129