Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic, space-independent coefficients, with an application to electrodiffusion

Bengt Nagel Division of Mathematical Physics, Royal Institute of Technology

TBD mathscidoc: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.
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@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.
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