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

Summary We study a Markovian process, the state space of which is the product of a set ofn points and the realx-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 inx, but in general time-inhomogeneous, this set of equations is studied with the help of the Fourier transformation. The marginal distribution in then discrete states corresponds to a time-inhomogeneousn-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 inx-space, with mean value and variance proportional tot. Explicit expressions for the proportionality factors are derived. The general theory is applied to the electrodiffusion equations, corresponding ton=2.