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arxiv: 2605.18136 · v1 · pith:PLBNSRFHnew · submitted 2026-05-18 · 🧮 math.PR

L\'evy processes with partially stochastic resetting

Zbigniew Palmowski , Noah Beelders , Lewis Ramsden , Apostolos D. Papaioannou This is my paper

Pith reviewed 2026-05-20 00:45 UTC · model grok-4.3

classification 🧮 math.PR
keywords Lévy processesstochastic resettingscale functionsexit problemsresolvent seriesintegral equationsfluctuation theory
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The pith

Lévy processes with proportional position resets at Poisson times have exit problems solved using a new family of scale functions derived from resolvent series.

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

The paper shows how to solve exit problems for Lévy processes that experience proportional resets at Poisson arrival times. These resets introduce additional jumps proportional to the current position, downward when positive and upward when negative. The solutions for exit times and probabilities are expressed through a newly constructed family of scale functions. These functions arise by converting the Laplace transforms of the exit times into integral equations that are solved explicitly via resolvent series. The approach also includes an SDE representation of the process and a discussion of its existence and uniqueness.

Core claim

All identities for the exit problems are given in terms of a new family of scale functions. To obtain this family, the Laplace transform of the exit times is reduced to integral equations that are solved in terms of resolvent series.

What carries the argument

New family of scale functions obtained by solving integral equations for the Laplace transforms of exit times via resolvent series.

If this is right

  • Exit probabilities and expected exit times admit explicit expressions in terms of the new scale functions.
  • The process has an SDE representation whose strong existence and uniqueness hold under the stated conditions.
  • The same scale-function approach covers both positive and negative domains with the corresponding direction-dependent proportional jumps.
  • Classical fluctuation identities for Lévy processes extend directly to this resetting case.

Where Pith is reading between the lines

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

  • The resolvent-series method may extend to other state-dependent jump mechanisms in Lévy processes.
  • Numerical truncation of the resolvent series could yield practical approximations for exit quantities.
  • Analogous integral-equation reductions might apply to fluctuation problems with random-time interventions.

Load-bearing premise

The Laplace transform of the exit times for the resetting process reduces to integral equations that admit solutions via resolvent series without additional regularity conditions on the Lévy measure or the resetting rate.

What would settle it

A specific Lévy measure and resetting rate where the integral equation for an exit-time Laplace transform has no resolvent-series solution, or where the derived scale functions fail to recover the correct exit identities.

Figures

Figures reproduced from arXiv: 2605.18136 by Apostolos D. Papaioannou, Lewis Ramsden, Noah Beelders, Zbigniew Palmowski.

Figure 1
Figure 1. Figure 1: Spectrally negative Lévy process with resetting with p=0.6 The main aim of the paper is to develop fluctuation theory for the PSR-LP, which is achieved by construction a new class of scale functions (based on potential measures rather the classical scale functions) that fits the PSR-LP set up. Within this framework, the paper addresses two main difficulties arising in the partial resetting or AIMD processe… view at source ↗
read the original abstract

In this paper, we solve exit problems for a L\'evy process that resets proportionally to its current position at independent Poisson epochs times. This resetting causes an additional (proportional to its current level) downward (upward) jump when the current position of the process is on the positive (negative) domain. Such a process can be expressed as an SDE, whose existence and uniqueness it discussed. All identities are given in terms of new family of scale functions. To obtain the new family of scale functions, we reduce the problem of the LT of the exit times into integral equations that are solve in terms of resolvent series.

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 studies Lévy processes with partially stochastic resetting, in which the process experiences position-proportional downward (upward) jumps at Poisson epochs when in the positive (negative) half-line. The dynamics are expressed as an SDE whose existence and uniqueness are discussed. Exit problems are solved via a new family of scale functions obtained by reducing the Laplace transform of exit times to integral equations that are solved by means of resolvent series.

Significance. If the reduction and series construction are valid, the work extends classical scale-function theory for Lévy processes to a resetting setting and supplies a constructive, iterative method for the new scale functions. This could be useful for exit-time calculations in models with state-dependent resets, provided the requisite regularity conditions are verified.

major comments (2)
  1. [Abstract] Abstract (paragraph on scale functions): the reduction of the Laplace transform of exit times to integral equations solved via resolvent series is asserted without any statement of the regularity conditions (local integrability of the Lévy density near zero, decay at infinity, or bounds on the resetting intensity) needed to guarantee that the spectral radius of the integral operator is less than one and that the series converges to the correct scale functions. This step is load-bearing for the claimed identities.
  2. [Abstract] Abstract: the existence and uniqueness of the SDE is mentioned but no reference is given to the precise conditions on the Lévy measure and the resetting rate that ensure the strong solution exists; this underpins the subsequent integral-equation analysis.
minor comments (2)
  1. [Abstract] Typo: 'solve' should read 'solved'.
  2. [Abstract] The abstract states that 'all identities are given in terms of new family of scale functions' yet does not indicate how these functions relate to or reduce to the classical scale functions when the resetting rate vanishes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating where revisions will be incorporated to enhance the rigor and clarity of the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on scale functions): the reduction of the Laplace transform of exit times to integral equations solved via resolvent series is asserted without any statement of the regularity conditions (local integrability of the Lévy density near zero, decay at infinity, or bounds on the resetting intensity) needed to guarantee that the spectral radius of the integral operator is less than one and that the series converges to the correct scale functions. This step is load-bearing for the claimed identities.

    Authors: We agree that an explicit statement of the regularity conditions is needed in the abstract to support the convergence claim. The manuscript works under the standard Lévy process assumptions (Lévy density locally integrable near zero with suitable decay at infinity) together with a bounded resetting intensity. To address this, we will revise the abstract and add a short preliminary paragraph specifying that these conditions ensure the spectral radius of the integral operator is strictly less than one, thereby justifying the resolvent series representation of the scale functions. revision: yes

  2. Referee: [Abstract] Abstract: the existence and uniqueness of the SDE is mentioned but no reference is given to the precise conditions on the Lévy measure and the resetting rate that ensure the strong solution exists; this underpins the subsequent integral-equation analysis.

    Authors: The body of the manuscript discusses existence and uniqueness of the SDE under the usual integrability condition on the Lévy measure and a positive finite resetting rate. We acknowledge that the abstract omits a reference to the supporting literature. In the revision we will insert a brief reference to standard results on SDEs with jumps (e.g., Applebaum, Lévy Processes and Stochastic Calculus) and explicitly restate the precise conditions on the Lévy measure and resetting intensity already used in the analysis. revision: yes

Circularity Check

0 steps flagged

Derivation from SDE to resolvent series is constructive and self-contained

full rationale

The paper defines the resetting Lévy process via an SDE whose existence and uniqueness are discussed, then reduces the Laplace transform of exit times to integral equations solved by resolvent series to produce the new scale functions. No quoted step shows a target quantity defined in terms of itself, a fitted parameter renamed as a prediction, or a central claim resting solely on an unverified self-citation. The method is a direct application of resolvent techniques to the derived integral equations, making the derivation independent of its outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard properties of Lévy processes and the well-posedness of the SDE with resetting; the central contribution is the new scale functions derived from the model rather than additional free parameters or invented physical entities.

axioms (2)
  • standard math Lévy processes possess stationary and independent increments and are stochastically continuous.
    Invoked implicitly when defining the base process before adding the resetting mechanism.
  • domain assumption Resetting occurs at the epochs of an independent Poisson process with constant intensity.
    Stated in the model description as the timing mechanism for proportional jumps.
invented entities (1)
  • New family of scale functions for the resetting process no independent evidence
    purpose: Expressing Laplace transforms of exit times and related identities
    Introduced to solve the exit problems; no external empirical or theoretical validation outside the derivation is mentioned.

pith-pipeline@v0.9.0 · 5638 in / 1604 out tokens · 55282 ms · 2026-05-20T00:45:21.817633+00:00 · methodology

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