Harmonic problems arising from continuous time random walks limit processes

2025/05/20 by Biočić, Ivan, Toaldo, Bruno
#31C05 #35R11 #60K15 #60K50 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2505.14550

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.

Citations

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Related

• Limit theorems for continuous time random walks with continuous paths
• Lagging/Leading Coupled Continuous Time Random Walks, Renewal Times and their Joint Limits
• Modelling Anomalous Diffusion: The Role of CTRWs and Non-Local Dynamics
• Space-time coupled evolution equations and their stochastic solutions
• A Universal Algorithm for Continuous Time Random Walks Limit Distributions