Feller property and convergence for semigroups of time-changed processes

TL;DR

This paper introduces a new approach to the Feller property for semigroups of time-changed processes, providing conditions for Feller property and establishing convergence results for associated semigroups and processes.

Contribution

It offers a novel substitute for the Feller property and new criteria for semigroup Feller-ness, along with convergence results for time-changed processes.

Findings

Established new conditions for Feller property of time-changed semigroups.

Proved convergence of semigroups and resolvents under measure convergence.

Demonstrated applications to evolution equations and process convergence.

Abstract

We give a substitute to Feller property for semigroups of time-changed processes; under some conditions this leads to establish sufficient (new) conditions for the semigroups to be Feller. Moreover, given a standard process and a sequence of measures converging vaguely to a final measure, under some assumptions, we establish convergence of the sequence of the semigroups and the resolvents of the corresponding time changed-processes. Some applications are given: convergence of solutions of evolution equations and convergence of finite time distributions, as well as weak convergence of the related processes.