$p$-Adic Diffusion and Random Walks on $[0, 1]$

Abstract

Integral operators on the real unit interval are constructed as transported from ones on the $p$-adic unit disc via the Monna map. This gives rise to strong Markov processes on $[0, 1]$, whose paths are right-continuous and have no discontinuities other than jumps. The spectra of the corresponding diffusion operators, whose kernel functions depend on a $p$-adic distance, are calculated. Further, the solution to the Cauchy problem for their heat equations are approximated via continuous- time random walks on finite sets coming from a hierarchical partition of the unit interval induced by the $p$-adic distance. The transport of $p$-adic diffusion to the real domain via the Monna map gives rise to a simple visualisation method. Illustrations of concrete examples are undertaken in the end.