Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups

TL;DR

This paper explores subordinators and their inverse processes, deriving governing equations using convolution-type operators, and examines how time-changing $C_0$-semigroups with hitting-times affects their properties and introduces non-local operators with long-range dependence.

Contribution

It introduces a general framework for governing equations of subordinators and their inverses using convolution-type operators, and analyzes the impact of hitting-time based time-changes on $C_0$-semigroups.

Findings

Hitting-times lead to non-local integro-differential operators.

Time-changed $C_0$-semigroups lose the semigroup property.

Long-range dependence is present in the operators studied.

Abstract

In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence.