Time-inhomogeneous jump processes and variable order operators

TL;DR

This paper introduces a class of non-decreasing, time-inhomogeneous jump processes that generalize subordinators, leading to new two-parameter semigroups and variable order fractional equations with applications to generalized subordinate Brownian motion.

Contribution

It proposes a novel class of jump processes with non-homogeneous increments, extending subordinators and developing associated semigroups and fractional equations.

Findings

Defined non-decreasing, time-inhomogeneous jump processes.

Derived time-dependent generators and Phillips formula.

Applied to generalized subordinate Brownian motion.

Abstract

In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not L\'evy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bern\v{s}tein functions for each time $t$. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined.