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arxiv: 2605.29030 · v1 · pith:KDKQVEED · submitted 2026-05-27 · math.PR

Preferential relocations enhance survival for Markov chains with killing

Jean Bertoin , Martin Minchev This is my paper

pith:KDKQVEEDreviewed 2026-06-29 09:57 UTCmodel grok-4.3open to challenge →

classification math.PR
keywords Markov chains with killingpersistence ratepreferential relocationssub-stochastic chainschains with complete connectionsFeynman-Kac formulasurvival probabilityergodic properties
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The pith

Preferential relocations to visited states slow the decay of survival probability in sub-stochastic Markov chains

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

The paper establishes that modifying a sub-stochastic Markov chain to include random relocations back to previously visited states raises its persistence rate, so the probability of still being alive at large times falls more slowly than in the unmodified chain. This strict improvement holds when the chain meets standard ergodicity conditions for chains with complete connections. The authors obtain explicit lower bounds on the ratio of the two persistence rates when the relocation measure is sufficiently spread out. A reader might care because the result quantifies how memory of past locations can counteract killing in stochastic models of survival or absorption.

Core claim

Our central result is that such preferential relocations increase the persistence rate, meaning the survival probability decays more slowly than for the benchmark chain without relocations, and this improvement is strict under mild assumptions. We derive explicit lower bounds on the ratio of persistence rates when the relocation distribution is highly dispersed. The analysis relies on ergodic properties of so-called chains with complete connections, and a Feynman-Kac approach to estimate persistence.

What carries the argument

The relocation mechanism that redirects the chain to a previously visited state drawn from a fixed distribution, whose effect on the killed process is bounded via ergodicity of the associated chain with complete connections.

If this is right

  • The persistence rate is strictly larger once relocations are added, under the stated ergodicity.
  • Explicit quantitative lower bounds on the ratio of rates are available when the relocation distribution has high dispersion.
  • The Feynman-Kac representation remains valid for the modified killed chain and yields the comparison.
  • The improvement applies to any initial distribution once the ergodic assumptions hold.
  • The result is new for the class of chains with killing that admit such relocations.

Where Pith is reading between the lines

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

  • The same relocation idea could be tested numerically on finite-state killed chains to measure the actual ratio of decay rates.
  • If the relocation distribution is concentrated rather than dispersed, the improvement may become arbitrarily small but still positive.
  • The technique might extend to continuous-state processes such as diffusions with killing by replacing the discrete relocation step with a jump to a previously visited location sampled from the occupation measure.
  • The bounds could be used to compare different relocation kernels and select one that maximises the guaranteed improvement.
  • Applications to population models with local extinction might treat relocation as a form of dispersal that revisits occupied patches.

Load-bearing premise

The underlying chain must satisfy the ergodic properties of chains with complete connections that deliver the strict improvement and the explicit bounds.

What would settle it

A concrete sub-stochastic chain satisfying the complete-connection ergodicity conditions for which the survival probability with relocations decays at least as fast as without them would falsify the strict improvement.

Figures

Figures reproduced from arXiv: 2605.29030 by Jean Bertoin, Martin Minchev.

Figure 1
Figure 1. Figure 1: Concentration of the occupation measure θε(0) for geometric relocation laws τε(k) = ε(1 − ε) k , for six decreasing values of ε. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical logarithmic comparison of r(ε), the benchmark value r, and the asymptotic lower bound from Corollary 5.2, for geometric relocation laws τε(k) = ε(1 − ε) k . It is remarkable that the lower bound for the limit inferior of r(ε) given there appears to be the exact value of the limit. Moreover, r(ε) seems always to lie between the lower bound r and this limit. We do not have a satisfactory explanatio… view at source ↗
read the original abstract

We investigate the impact on survival of a modification of the evolution of a sub-stochastic Markov chain that involves random relocations at previously visited states. Our central result is that such preferential relocations increase the persistence rate, meaning the survival probability decays more slowly than for the benchmark chain without relocations, and this improvement is strict under mild assumptions. We derive explicit lower bounds on the ratio of persistence rates when the relocation distribution is highly dispersed. The analysis relies on ergodic properties of so-called chains with complete connections, and a Feynman--Kac approach to estimate persistence.

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

2 major / 2 minor

Summary. The paper claims that adding preferential random relocations to previously visited states in a sub-stochastic Markov chain with killing strictly increases the persistence rate (i.e., the survival probability decays more slowly) relative to the unmodified chain, under mild assumptions. Explicit lower bounds on the ratio of the two persistence rates are derived when the relocation distribution is highly dispersed. The argument relies on showing that the modified process is a chain with complete connections whose ergodic properties, combined with a Feynman-Kac representation, yield the strict improvement.

Significance. If the central claim holds, the result supplies a concrete mechanism by which memory encoded in relocation preferences can improve long-term survival in killed processes, together with quantitative bounds. The explicit use of complete-connection ergodic theory and the Feynman-Kac approach, when the required positivity conditions are verified, would constitute a technically clean contribution to the literature on persistence and quasi-stationary distributions.

major comments (2)
  1. [Main theorem and the ergodic argument] The strict inequality asserted in the main result rests on the claim that the relocation mechanism automatically produces a chain with complete connections satisfying the uniform lower bounds on cylinder probabilities needed for the ergodic comparison. The manuscript must exhibit the precise interaction between the original sub-stochastic kernel and the relocation measure that guarantees these bounds; without an explicit verification or additional hypothesis on the support of the relocation distribution, the strict improvement may fail for some natural choices (e.g., concentrated relocation measures).
  2. [Feynman-Kac section] The Feynman-Kac representation used to compare persistence rates must be shown to remain valid after the relocation modification; in particular, the change of measure induced by the relocations should not alter the killing structure in a way that invalidates the representation or the subsequent application of the ergodic theorem.
minor comments (2)
  1. [Abstract] The abstract refers to 'mild assumptions' without listing them; a short explicit list in the introduction would improve readability.
  2. [Notation and definitions] Notation for the persistence rate (or its logarithm) should be introduced once and used consistently when stating the ratio bounds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will revise the manuscript accordingly to provide the requested explicit verifications while preserving the original claims under the stated mild assumptions.

read point-by-point responses

  1. Referee: [Main theorem and the ergodic argument] The strict inequality asserted in the main result rests on the claim that the relocation mechanism automatically produces a chain with complete connections satisfying the uniform lower bounds on cylinder probabilities needed for the ergodic comparison. The manuscript must exhibit the precise interaction between the original sub-stochastic kernel and the relocation measure that guarantees these bounds; without an explicit verification or additional hypothesis on the support of the relocation distribution, the strict improvement may fail for some natural choices (e.g., concentrated relocation measures).

    Authors: The central theorem already incorporates a mild dispersion hypothesis on the relocation measure precisely to guarantee the uniform lower bounds on cylinder probabilities through its interaction with the positive transitions of the original sub-stochastic kernel. To address the request for explicit verification, we will insert a new lemma that derives the cylinder lower bound explicitly as a product of the kernel's minimal transition probability and the dispersion parameter. This also makes clear that concentrated relocation measures fall outside the hypotheses and may indeed fail to produce strict improvement, consistent with the theorem statement. revision: yes

  2. Referee: [Feynman-Kac section] The Feynman-Kac representation used to compare persistence rates must be shown to remain valid after the relocation modification; in particular, the change of measure induced by the relocations should not alter the killing structure in a way that invalidates the representation or the subsequent application of the ergodic theorem.

    Authors: The killing structure is encoded solely in the mass defect of the original sub-stochastic kernel and is unaffected by the relocation step, which only redistributes probability mass among visited states. The Feynman-Kac representation therefore applies verbatim to the modified kernel, and the ergodic theorem for chains with complete connections directly yields the comparison of persistence rates. We will add a short verification paragraph in the Feynman-Kac section confirming that the representation and subsequent ergodic application remain valid without alteration to the killing mechanism. revision: yes

Circularity Check

0 steps flagged

No circularity: central claim rests on external ergodic theory and Feynman-Kac estimates, not self-definition or fitted inputs

full rationale

The abstract states that the strict improvement in persistence rate follows from ergodic properties of chains with complete connections together with a Feynman-Kac representation. These are invoked as standard tools rather than derived within the paper or reduced to the relocation mechanism by construction. No equations are presented that equate the persistence-rate ratio to a fitted parameter or to a self-citation chain. The 'mild assumptions' are external conditions on the induced chain, not tautological definitions. The derivation chain therefore remains independent of its target conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no free parameters, invented entities, or additional axioms can be identified.

axioms (2)
  • domain assumption Ergodic properties of chains with complete connections hold for the processes under study
    Invoked in the final sentence of the abstract to support the analysis.
  • domain assumption Feynman-Kac representation applies to estimate persistence
    Cited as the method used to estimate persistence.

pith-pipeline@v0.9.1-grok · 5611 in / 1183 out tokens · 25034 ms · 2026-06-29T09:57:38.688483+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages

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