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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We prove that the radii of convergence of the solutions of a$p$-adic differential equation $${\fancyscript{F}}$$ over an affinoid domain$X$of the Berkovich affine line are continuous functions on$X$that factorize through the retraction of $${X\to\Gamma}$$ of$X$onto a finite graph $${\Gamma\subseteq X}$$ . We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on$X$is controlled by a$finite$family of data.

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