Stochastic processes and their spectral representations over non-archimedean fields

TL;DR

This paper explores stochastic processes over non-archimedean fields, focusing on their spectral representations, stochastic integrals, and vector-valued measures in infinite-dimensional spaces.

Contribution

It introduces spectral decomposition theorems and studies stochastic integrals and measures in non-archimedean vector spaces, expanding the theoretical framework.

Findings

Spectral decompositions for non-archimedean stochastic processes proved.

Development of stochastic integrals in non-archimedean vector spaces.

Analysis of vector-valued measures over non-archimedean fields.

Abstract

The article is devoted to stochastic processes with values in finite- and infinite-dimensional vector spaces over infinite fields $\bf K$ of zero characteristics with non-trivial non-archimedean norms. For different types of stochastic processes controlled by measures with values in $\bf K$ and in complete topological vector spaces over $\bf K$ stochastic integrals are investigated. Vector valued measures and integrals in spaces over $\bf K$ are studied. Theorems about spectral decompositions of non-archimedean stochastic processes are proved.