## Journal of the Mathematical Society of Japan

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

#### Abstract

We have developed the theory of $\mathrm{KM}_{2O}$-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 $\mathrm{KM}_{2O}$-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 $\mathrm{KM}_{2O}-$ 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

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

Digital Object Identifier
doi:10.2969/jmsj/1191419129

Mathematical Reviews number (MathSciNet)
MR1961299

Zentralblatt MATH identifier
1162.82312

#### 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