Spiking and Resetting
C\'edric Bernardin, Vsevolod Vladimirovich Tarsamaev

TL;DR
This paper rigorously analyzes a PDMP with resetting, showing that as a small parameter vanishes, the associated point process converges to a jump Markov process with spikes modeled by a Poisson point process, confirming previous conjectures.
Contribution
It provides a rigorous proof of the convergence of the resetting process to a decorated jump Markov process in the singular limit, confirming earlier heuristic results.
Findings
Convergence of the resetting process to a jump Markov process as epsilon approaches zero.
Characterization of spikes as a Poisson point process with specific intensity.
Validation of previous conjectures regarding the limiting behavior of the PDMP.
Abstract
We consider a one-dimensional piecewise deterministic Markov process (PDMP) on with resetting at and depending on a small parameter . In the singular vanishing limit we prove that the `` resetting '' simple point process associated to the PDMP converges to a point process described by a jump Markov process decorated by ``spikes'' distributed as a time-space Poisson point process with intensity proportional to . This proves rigorously results appeared previously in \cite{SBDKC25} and also justifies partially a conjecture formulated there.
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Taxonomy
TopicsDiffusion and Search Dynamics · Distributed Control Multi-Agent Systems · stochastic dynamics and bifurcation
