J. -P. KahaneUniversity of Montpellier and Washington University, St. Louis, MissouriMary WeissUniversity of Montpellier and Washington University, St. Louis, MissouriGuido WeissUniversity of Montpellier and Washington University, St. Louis, Missouri
We study a Markovian process, the state space of which is the product of a set of$n$points and the real$x$-axids. Under certain regularity conditions this study is equivalent to investigating the solution of a set of couple diffusion equations, generalization of the Fokker-Planck (or second Kolmogorov) equation. Assuming the process homogeneous in$x$, but in general time-inhomogeneous, this set of equations is studied with the help of the Fourier transformation. The marginal distribution in the$n$discrete states corresponds to a time-inhomogeneous$n$-state Markov chain in continuous time. The properties of such a Markov chain are studied, especially the asymptotic behaviour in the time-periodic case. We obtain a natural generalization of the well-known asymptotic behaviour in the time-homogeneous case, finding a subdivision of the states into groups of essential states, the distribution inside easch group being asymptotically periodic and independent of the starting distribution. Next, still assuming time-periodicity, we study the asymptotic behaviour of the complete Markovian process, showing that inside each of the groups mentioned above the distribution approaches a common normal distribution in$x$-space, with mean value and variance proportional to$t$. Explicit expressions for the proportionality factors are derived.