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Spiking and Resetting

Cédric Bernardin, Vsevolod Vladimirovich Tarsamaev

TL;DR

This work rigorously derives the singular spiking limit for a one-dimensional PDMP with resetting, showing that the time-space pre-spike points converge to a decorated Poisson point process with intensity f(0)^2 dt ⊗ x^{-2} dx. The authors develop a robust renewal- and Laplace-transform framework, proving conditional no-jump rectangle convergence and then full Poisson convergence, while providing explicit asymptotics for key generating-function components E^ε, D^ε, and C^ε. These results vindicate conjectures from prior physics literature and clarify the mechanism by which spiking events emerge in the small-ε limit. The work highlights a deep connection between resetting dynamics, boundary-layer behavior, and Poissonian spiking structures that may inform broader resetting and PDMP analyses.

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.

Spiking and Resetting

TL;DR

This work rigorously derives the singular spiking limit for a one-dimensional PDMP with resetting, showing that the time-space pre-spike points converge to a decorated Poisson point process with intensity f(0)^2 dt ⊗ x^{-2} dx. The authors develop a robust renewal- and Laplace-transform framework, proving conditional no-jump rectangle convergence and then full Poisson convergence, while providing explicit asymptotics for key generating-function components E^ε, D^ε, and C^ε. These results vindicate conjectures from prior physics literature and clarify the mechanism by which spiking events emerge in the small-ε limit. The work highlights a deep connection between resetting dynamics, boundary-layer behavior, and Poissonian spiking structures that may inform broader resetting and PDMP analyses.

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.
Paper Structure (10 sections, 19 theorems, 216 equations, 2 figures)

This paper contains 10 sections, 19 theorems, 216 equations, 2 figures.

Table of Contents

  1. Introduction, Model and Result
  2. Rectangle convergence conditionally to no-jump
  3. Convergence to a Poisson process: proof of Theorem \ref{['thm:mainone']}
  4. Proof of Proposition \ref{['prop:nj-laplace-generating-convergence']}
  5. Expansion of $E^\varepsilon (\sigma)$
  6. Expansion of $D^\varepsilon (\sigma)$
  7. Expansion of $C^\varepsilon (\sigma)$
  8. Probabilistic technical lemmas
  9. Asymptotic of integrals and consequences
  10. Uniform convergence from Laplace transform convergence

Key Result

Theorem 1.1

Let us fix $\delta \in (0,1]$ and $T>0$. We equip the space of Borel measures on the compact set $[0,T] \times [\delta, 1]$ with the weak topology. We have that the restriction of the simple point process $(\mathbb Q^\varepsilon)_{\varepsilon>0}$ to $[0,T] \times [\delta, 1]$ converges in law to the

Figures (2)

  • Figure 1: A formal realisation of the PDMP $(X_t^\varepsilon)_{t \ge 0}$.
  • Figure 2: A formal realisation of the PDMP $({\tilde{X}}_t^\varepsilon)_{t \ge 0}$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • ...and 27 more