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arxiv: 2605.00148 · v1 · submitted 2026-04-30 · 🧮 math.PR · math-ph· math.MP

Persistence in perturbed contact models in continuum

S. Pirogov , E Zhizhina This is my paper

Pith reviewed 2026-05-09 20:29 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords contact processesinvariant measuresFeynman-Kac formulacorrelation functionsnonlocal operatorscontinuum modelspersistence
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The pith

Local peaks in mortality do not lead to extinction in perturbed contact models on metric spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends prior results on contact processes by removing the requirement for a critical balance between birth and death rates. Instead, it shows that the correlation functions evolve according to a nonlocal convolution operator perturbed by a negative potential. Under these conditions, a family of invariant measures still exists and can be expressed using the Feynman-Kac formula. This demonstrates that the models remain persistent, with populations not dying out due to localized high death rates.

Core claim

When the balance condition between birth and death rates is violated, the evolution of correlation functions is determined by a nonlocal convolution type operator perturbed by a negative potential. This setup admits a family of invariant measures that can be described using the Feynman-Kac formula, implying that local peaks in mortality do not typically lead to extinction of the process.

What carries the argument

The nonlocal convolution-type operator perturbed by a negative potential, which governs the dynamics of the correlation functions and allows application of the Feynman-Kac formula to construct invariant measures.

If this is right

  • The contact process persists despite violations of the criticality condition.
  • Invariant measures can be constructed explicitly via the Feynman-Kac representation.
  • Local disasters in mortality do not cause global extinction in these models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework may apply to other interacting particle systems where potentials perturb the base operator.
  • Similar techniques could be used to study long-time behavior in non-critical regimes of branching processes on spaces.
  • Testing the formula on specific metric spaces like Euclidean space could reveal explicit forms for the measures.

Load-bearing premise

The rates satisfy conditions such that the evolution of correlation functions is governed by a nonlocal convolution-type operator perturbed by a negative potential, without the critical regime balance between birth and death.

What would settle it

A counterexample where the Feynman-Kac constructed measures fail to be invariant for the process under the perturbed operator would disprove the existence claim.

read the original abstract

Can a local disaster lead to extinction? We answer this question in this work. In the paper \cite{PZ-PPI} we considered contact processes on locally compact metric spaces with state dependent birth and death rates and formulated sufficient conditions on the rates that ensure the existence of invariant measures. One of the crucial conditions in \cite{PZ-PPI} was the critical regime condition, which meant the existence of a balance between birth and death rates in average. In the present work, we reject the criticality condition and suppose that the balance condition is violated. This implies that the evolution of the correlation functions of the contact model under consideration is determined by a nonlocal convolution type operator perturbed by a (negative) potential. We show that local peaks in mortality do not typically lead to extinction. We prove that a family of invariant measures exists even without the criticality condition and these measures can be described using the Feynman-Kac formula.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript extends the authors' earlier work on contact processes on locally compact metric spaces with state-dependent birth and death rates by dropping the criticality condition (average balance between birth and death). It shows that the correlation functions evolve according to a nonlocal convolution-type operator perturbed by a negative potential, establishes that this perturbation lies in the Kato class so that the multiplicative functional is well-defined, and uses the Feynman-Kac formula to construct a family of invariant measures. The existence of these measures is then invoked to conclude that local peaks in mortality do not typically lead to extinction.

Significance. If the stated conditions on the rates hold and the semigroup generation plus Kato-class perturbation are verified, the result supplies a natural generalization of persistence theorems beyond the critical regime. The explicit Feynman-Kac representation of the invariant measures is a concrete strength that may enable further quantitative analysis of long-term behavior in continuum contact models.

minor comments (2)
  1. [Introduction] The introduction would benefit from a short paragraph explicitly listing the rate conditions that guarantee the unperturbed operator generates a positive semigroup (currently referenced only in passing).
  2. Notation for the perturbed generator (e.g., the precise definition of the negative potential term) should be introduced once in a dedicated display equation before its repeated use in the Feynman-Kac construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The referee's summary correctly identifies the main contributions: the removal of the criticality condition from our earlier work, the representation of the correlation-function evolution via a nonlocal operator perturbed by a negative potential, verification that the perturbation belongs to the Kato class, and the construction of invariant measures via the Feynman-Kac formula. We appreciate the recommendation for minor revision and will prepare a revised version accordingly.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends prior work by explicitly dropping the criticality condition from PZ-PPI and deriving existence of invariant measures for the non-critical regime. It states conditions on rates that make the unperturbed nonlocal convolution operator generate a positive semigroup, places the negative potential in the Kato class, and applies the Feynman-Kac formula to obtain the measures. These steps are presented as new analysis rather than a re-derivation or fit of the prior critical case; no equation or claim reduces by construction to a self-citation, fitted parameter, or renamed input. The central persistence result therefore rests on independent semigroup and perturbation arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on unspecified conditions on the birth and death rates that make the correlation evolution operator well-defined as a perturbed convolution.

axioms (1)
  • domain assumption The birth and death rates satisfy conditions allowing the correlation functions to evolve according to a nonlocal convolution operator perturbed by a negative potential.
    Stated as the mechanism determining the evolution when the criticality condition is dropped.

pith-pipeline@v0.9.0 · 5451 in / 1210 out tokens · 27241 ms · 2026-05-09T20:29:53.608327+00:00 · methodology

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Reference graph

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