Non-Archimedean pseudodifferential operators and Feller Semigroups

TL;DR

This paper investigates non-Archimedean pseudodifferential operators with negative definite symbols, establishing their role as generators of Feller semigroups and introducing new anisotropic Sobolev spaces for their analysis.

Contribution

It introduces a new class of anisotropic Sobolev spaces and proves these pseudodifferential operators generate Feller semigroups, advancing the understanding of non-Archimedean analysis.

Findings

Operators extend to generators of Feller semigroups

Introduction of anisotropic Sobolev spaces as natural domains

Analysis of the Cauchy problem for pseudodifferential equations

Abstract

In this article we study a class of non-Archimedean pseudodifferential operators whose symbols are negative definite functions. We prove that these operators extend to generators of Feller semigroups. In order to study these operators, we introduce a new class of anisotropic Sobolev spaces, which are the natural domains for the operators considered here. We also study the Cauchy problem for certain pseudodifferential equations.