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A random walk on \(p\)-adics – the generator and its spectrum. (English) Zbl 0810.60065

This is a continuation of [authors, Diffusion on \(p\)-adic numbers. In: K. Itô and T. Hida (eds.), Gaussian random fields, Proc. Nagoya Conf. (Singapore, 1991)]. For a prime number \(p\), let \[ Q_ p = \left\{ x:x = \sum_{i=N}^ \infty \gamma_ i p^ i, \;N \in \mathbb{Z}, \gamma_ i = 0,1, \dots, p-1 \right\} \] be the set of \(p\)-adic numbers and equipped with the norm \(| x | = p^{-i_ 0}\), where \(i_ 0\) is the smallest \(i\) such that \(\gamma_ i \neq 0\). Then it makes \(Q_ p\) a complete separable metric locally compact totally disconnected space. A continuous time random walk on \(Q_ p\) is constructed, the core of its \(L^ 2\) infinitesimal \(-H\) is given explicitly and \(H\) is proven to have pure point spectrum. The corresponding Dirichlet form is shown of jump type.

MSC:

60G50 Sums of independent random variables; random walks
47B25 Linear symmetric and selfadjoint operators (unbounded)
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