# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332218

Arkiv for Matematik, 5, (3), 363-390, 1964.9
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.
```@inproceedings{bengt1964properties,
title={Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic, space-independent coefficients, with an application to electrodiffusion},
author={Bengt Nagel},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203432343920027},
booktitle={Arkiv for Matematik},
volume={5},
number={3},
pages={363-390},
year={1964},
}
```
Bengt Nagel. Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic, space-independent coefficients, with an application to electrodiffusion. 1964. Vol. 5. In Arkiv for Matematik. pp.363-390. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203432343920027.