Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields

TL;DR

This paper develops the theory of stochastic processes in non-Archimedean spaces, establishing foundational theorems and constructing classes of processes with measures valued in non-Archimedean fields.

Contribution

It introduces the non-Archimedean analogs of Kolmogorov, Lévy, and Poisson theorems, expanding stochastic process theory into non-Archimedean settings.

Findings

Proved the non-Archimedean Kolmogorov theorem.

Studied non-Archimedean Markov and Poisson processes.

Constructed wide classes of non-Archimedean stochastic processes.

Abstract

Stochastic processes on topological vector spaces over non-Archimedean fields and with transition measures having values in non-Archimedean fields are defined and investigated. For this the non-Archimedean analog of the Kolmogorov theorem is proved. The analogos of Markov and Poisson processes are studied. For Poisson processes the corresponding Poisson measures are considered and the non-Archimedean analog of the L\`evy theorem is proved. Wide classes of stochastic processes are constructed.