## On a KM$_2$O-Langevin equation} {\Large\bf with continuous time\ (1)

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 4, Page 545--593.

Okabe, Yasunori
On a KM$_2$O-Langevin equation} {\Large\bf with continuous time\ (1)
We treat a continuous time stationary Gaussian process $\mbox{\bf X}$ whose global time evolution is governed by $[\alpha,\beta,\gamma]$-Langevin equation and derive a KM$_2$O-Langevin equation which governs a local time evolution of the stochastic process $\mbox{\bf X}$. Moreover, we prove a fluctuation-dissipation theorem based upon the KM$_2$O-Langevin equation and derive a system of equations characterizing both the fluctuation and the drift coefficients in the KM$_2$O-Langevin equation from the covariance function of the stationary process $\mbox{\bf X}$.
Keywords: $[\alpha,\beta,\gamma]$-Langevin equation, KMO-Langevin equation with continuous time, KM$_2$O-Langevin equation with continuous time, Fluctuation-dissipation theorem.