## 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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