A strong Markov process time-changed by an inverse killed subordinator

TL;DR

This paper introduces a novel class of generalized time-fractional PDEs by analyzing Markov processes time-changed with inverse killed subordinators, extending previous fractional calculus models.

Contribution

It establishes a new link between Bernstein functions with infinite Lévy measure and generalized time-fractional PDEs, broadening the scope of probabilistic representations.

Findings

Established a correspondence between Bernstein functions and PDEs

Extended fractional equations to include inverse killed subordinators

Provided a probabilistic framework for generalized time-fractional PDEs

Abstract

In this paper, we consider a type of time-changed Markov process, where the time-change is an inverse killed subordinator. This can be seen as an extension of Chen (Chen, Z., Time fractional equations and probabilistic representation, Chaos Solitons and Fractals, 168-174, 2017). As a result, it constructs a one-to-one correspondence between general Bernstein functions (with infinite L\'{e}vy measure) and a class of generalized time-fractional partial differential equations.