Harmonic problems arising from continuous time random walks limit processes

Ivan Biočić; Bruno Toaldo

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arXiv · 2505.14550

Harmonic problems arising from continuous time random walks limit processes

Ivan Biočić·Bruno Toaldo

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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.

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Ivan Biočić, Bruno Toaldo. 2025-05-20. Harmonic problems arising from continuous time random walks limit processes. https://arxiv.org/abs/2505.14550

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