Sub-Feller Semigroups Generated by Pseudodifferential Operators on Symmetric Spaces of Noncompact Type

TL;DR

This paper extends the theory of pseudodifferential operators to symmetric spaces of noncompact type, establishing conditions under which these operators generate sub-Feller semigroups, thus generalizing Euclidean space results.

Contribution

It introduces a probabilistic symbol framework for pseudodifferential operators on symmetric spaces and characterizes their semigroup generation using the Hille-Yosida-Ray theorem.

Findings

Established conditions for pseudodifferential operators to generate sub-Feller semigroups.

Generalized Euclidean space results to symmetric spaces of noncompact type.

Connected probabilistic symbols with the characteristic exponent concept.

Abstract

We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions. The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's L\'evy-Khinchine formula to a function of two variables. The Hille-Yosida-Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space.