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

2026/08/03 by Patrick Erik Bradley, Paulina Halwas, Ell Ari Nitsche +1
Mathematics · #math.PR #msc:60B05 #msc:60G22 #msc:60J25

paper · pdf

23 pages, 4 figures

arxiv created 2026/08/03 · arxiv updated 2026/08/04

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.

Citations

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