Spiking and Resetting

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 $[0,1]$ with resetting at $0$ and depending on a small parameter $\varepsilon>0$. In the singular vanishing limit $\varepsilon \to 0$ 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 $dt \otimes x^{-2} dx$. This proves rigorously results appeared previously in \cite{SBDKC25} and also justifies partially a conjecture formulated there.