Harmonic problems arising from continuous time random walks limit processes

TL;DR

This paper introduces a universal method to identify non-local evolution equations for CTRW limit processes, establishing well-posedness in specific cases and unifying various approaches in the literature.

Contribution

The paper presents a new universal approach to derive and analyze non-local evolution equations for CTRW limit processes, including well-posedness results for specific cases.

Findings

Established well-posedness for Feller processes time-changed with overshooting of a subordinator

Unified multiple existing approaches to non-local evolution equations

Provided a method to identify governing equations for CTRW limit processes

Abstract

In this paper, we develop a universal method that identifies the (non-local) governing evolution equations for Continuous Time Random Walks' (CTRWs) limit processes. Given one of these processes, our method provides the form of a non-local operator, acting on space and time variables jointly, such that the (generalized) harmonic problem associated with it represents an evolution governing equation for this process. Then, the well-posedness of this problem must be established case by case. In this paper, we establish well-posedness when the process is a Feller process (on a general Polish space $E$) time-changed with the overshooting of a subordinator. Also, we will show how our method applies to several cases when the equation and its well-posedness are already known, hence unifying several different approaches in the literature.