## Arkiv för Matematik

- Ark. Mat.
- Volume 5, Number 3-4 (1964), 363-390.

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

#### Abstract

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.

The general theory is applied to the electrodiffusion equations, corresponding to *n*=2.

#### Article information

**Source**

Ark. Mat., Volume 5, Number 3-4 (1964), 363-390.

**Dates**

First available in Project Euclid: 31 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afm/1485893457

**Digital Object Identifier**

doi:10.1007/BF02591137

**Mathematical Reviews number (MathSciNet)**

MR220351

**Rights**

1965 © Swets & Zeitlinger B.V.

#### Citation

Nagel, Bengt. Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic, space-independent coefficients, with an application to electrodiffusion. Ark. Mat. 5 (1964), no. 3-4, 363--390. doi:10.1007/BF02591137. https://projecteuclid.org/euclid.afm/1485893457