Real time random walks on \(p\)-adic numbers (Q2712755)

The authors give a survey of various methods of constructing Markov processes over the field \(\mathbb Q_p\) of \(p\)-adic numbers. The most detailed exposition is given for the approach (developed by the authors, with a participation of Vilela Mendes and Zhao), in which first a special class of Markov chains on the set of balls of a fixed radius is constructed, in such a way that a chain tends to a process on \(\mathbb Q_p\), as the radius of the balls tends to zero. Properties of the resulting processes are studied. Some physical motivations are discussed. NEWLINENEWLINENEWLINEAmong other approaches there are those based on parabolic equations on \(\mathbb Q_p\), on the use of the Lévy-Khinchin representation etc. A process on the ``unit ball'', that is the ring of integers \(\mathbb Z_p\), is obtained from a process on \(\mathbb Q_p\) by considering \(\mathbb Z_p\) as a kind of a fundamental domain with respect to a certain discrete family of translations. Note that another construction of a process on \(\mathbb Z_p\) (obtained from a process on \(\mathbb Q_p\) by restricting possible jumps) was given by the reviewer [Methods Funct. Anal. Topol. 2, No. 3/4, 53-58 (1996; Zbl 0926.60091)]. There is also a theory of stochastic differential equations on \(\mathbb Q_p\) leading to a large class of Markov processes; see the reviewer [Potential Anal. 6, No. 2, 105-125 (1997; Zbl 0874.60047)].NEWLINENEWLINEFor the entire collection see [Zbl 0957.00063].