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Harmonic problems arising from continuous time random walks limit processes

Ivan Biočić, Bruno Toaldo

TL;DR

The paper develops a universal harmonic-operator framework to identify non-local, space-time evolution equations for CTRW limit processes, including time-changed Feller processes by overshoot of a subordinator. It constructs generalized harmonic problems with operators $\mathfrak{A}^+$ (and $\mathfrak{A}^-$ in the undershoot/alternate regime) and proves well-posedness for overshoot-driven limits, linking probabilistic hitting-time representations to evolution equations. The approach unifies a broad class of known fractional- and variable-order kinetic equations, and extends to space-dependent waiting times and coupled CTRW limits, providing a systematic method to derive governing equations. This framework has broad implications for modeling anomalous diffusion and non-local dynamics in heterogeneous media and complex time structures.

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.

Harmonic problems arising from continuous time random walks limit processes

TL;DR

The paper develops a universal harmonic-operator framework to identify non-local, space-time evolution equations for CTRW limit processes, including time-changed Feller processes by overshoot of a subordinator. It constructs generalized harmonic problems with operators (and in the undershoot/alternate regime) and proves well-posedness for overshoot-driven limits, linking probabilistic hitting-time representations to evolution equations. The approach unifies a broad class of known fractional- and variable-order kinetic equations, and extends to space-dependent waiting times and coupled CTRW limits, providing a systematic method to derive governing equations. This framework has broad implications for modeling anomalous diffusion and non-local dynamics in heterogeneous media and complex time structures.

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 ) 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.

Paper Structure

This paper contains 15 sections, 8 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Harmonic functions approach for time non-local equations
  3. Scaling limits of CTRWs and time-changed Markov processes
  4. Governing equations and generalized harmonicity
  5. Harmonicity problem for Markov additive $(A_u,\sigma_u)$
  6. Overshooted processes and coupled non-local operators
  7. The governing equation of the overshooted process
  8. Proof of Theorem \ref{['thm:operator']}
  9. Harmonic approach in action
  10. Uncoupled CTRWs and fractional kinetic equations
  11. Space-dependent CTRWs and variable-order fractional kinetic equations
  12. The undershooted process and coupled equations
  13. Proofs of auxiliary results
  14. On the core of (subordinated) Feller processes
  15. Proof of Lemma \ref{['l:aux-G-q(t)']}

Key Result

Theorem 2.1

Let $(A_u, \sigma_u)_u$ be a Feller process as above, with the transition probabilities $p_{u}(x,v;dy,dw)$. Then, for any $t\ge0$, there exist two Markov processes $\,^tZ^+=(^tA_u^+, \,^t\sigma_u^+)_u$ and $\,^tZ^-=(^tA_u^-, \,^t\sigma_u^-)_u$, on the canonical probability space with measures $\,^t\ respectively, where $\,^t\upvarpi^+(x,t-v,dy)\coloneqq \mathds{P}^{(x,t-v)}\left( A_{L_t}\in dy, \s

Theorems & Definitions (15)

  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.3: Courrège-type form of $\mathfrak{A}^+$
  • Example 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 4.1
  • Theorem 4.2
  • ...and 5 more