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arxiv: 2411.13099 · v2 · submitted 2024-11-20 · 🧮 math.PR

Long time behavior of killed Feynman-Kac semigroups with singular Schr{\"o}dinger potentials

Arnaud Guillin (LMBP) , D I Lu , Boris Nectoux (LMBP) , Liming Wu (LMBP) This is my paper

Pith reviewed 2026-05-23 17:27 UTC · model grok-4.3

classification 🧮 math.PR
keywords Feynman-Kac semigroupssingular Schrödinger potentialslong time behaviorFeller kernelsPerron-FrobeniusLévy processeskinetic Langevin processinteracting particles
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The pith

Killed Feynman-Kac semigroups with singular Schrödinger potentials remain compact for Lévy and kinetic processes, yielding long-time convergence.

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

This paper aims to prove that killed Feynman-Kac semigroups associated with various processes from statistical physics, equipped with very general singular Schrödinger potentials, are compact and display specific long-time behavior. The processes include solutions to elliptic equations, Lévy processes, kinetic Langevin dynamics with locally Lipschitz gradients, and interacting Lévy particle systems. A reader would care because these models link to quantum mechanics and Schrödinger operators, and establishing their asymptotic properties allows reliable prediction of equilibrium behavior even with irregular potentials. The approach uses a general Perron-Frobenius theorem for Feller kernels and spectral radius bounds to handle the singularities without extra assumptions.

Core claim

We show that the killed Feynman-Kac semigroups for the listed processes with singular potentials are compact Feller kernels. Therefore the Perron-Frobenius theorem from 2020 applies, giving a principal eigenvalue and eigenfunction, and the long-time behavior follows with exponential convergence to the quasi-stationary distribution. The essential spectral radius is controlled using bounds for nonnegative kernels.

What carries the argument

The killed Feynman-Kac semigroup, which incorporates the killing by the singular potential and acts as a Feller kernel on the state space.

If this is right

  • The semigroups possess a spectral gap leading to exponential ergodicity.
  • Unique positive eigenfunction exists corresponding to the principal eigenvalue.
  • Results cover both local and nonlocal operators including fractional Laplacians.
  • Interacting systems maintain the compactness property under singular interactions.

Where Pith is reading between the lines

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

  • This suggests the methods could handle even rougher potentials encountered in certain quantum models.
  • Applications might include improved analysis of ground states for Schrödinger operators with singularities.
  • Extensions to time-inhomogeneous cases or other Markov processes could be explored based on the same Feller property.

Load-bearing premise

The generated killed operators must qualify as Feller kernels for the Perron-Frobenius theorem to apply directly to these singular cases.

What would settle it

Constructing a specific singular potential and process, such as a Lévy process with a highly singular potential, where the semigroup operator is not compact or the essential spectral radius equals the spectral radius would falsify the result.

read the original abstract

In this work, we investigate the compactness and the long time behavior of killed Feynman-Kac semigroups of various processes arising from statistical physics with very general singular Schr{\"o}dinger potentials. The processes we consider cover a large class of processes used in statistical physics, with strong links with quantum mechanics and (local or not) Schr{\"o}dinger operators (including e.g. fractional Laplacians). For instance we consider solutions to elliptic differential equations, L{\'e}vy processes, the kinetic Langevin process with locally Lipschitz gradient fields, and systems of interacting L{\'e}vy particles. Our analysis relies on a Perron-Frobenius type theorem derived in a previous work [A. Guillin, B. Nectoux, L. Wu, 2020 J. Eur. Math. Soc.] for Feller kernels and on the tools introduced in [L. Wu, 2004, Probab. Theory Relat. Fields] to compute bounds on the essential spectral radius of a bounded nonnegative kernel.

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 manuscript investigates the compactness and long-time behavior of killed Feynman-Kac semigroups generated by Lévy processes, kinetic Langevin dynamics with locally Lipschitz gradients, and systems of interacting Lévy particles, all subject to very general singular Schrödinger potentials. The analysis applies the Perron-Frobenius theorem from Guillin-Nectoux-Wu (2020 JEMS) for Feller kernels together with essential spectral radius bounds from Wu (2004) to obtain spectral gap and asymptotic results.

Significance. If the Feller property is verified for the killed kernels, the work would furnish a unified spectral framework for long-time asymptotics across several statistical-physics models linked to Schrödinger operators (including fractional Laplacians), extending the 2020 theorem to singular potentials without additional regularity. The explicit dependence on two prior results is a strength for breadth but limits standalone novelty.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the claim that the analysis 'relies on' the 2020 Perron-Frobenius theorem presupposes that the killed Feynman-Kac operators with singular V remain Feller kernels for all listed processes; no separate verification or additional assumptions ensuring continuity preservation after multiplication by the exponential functional of V are supplied, rendering the theorem's applicability unconfirmed and load-bearing for the compactness and long-time claims.
  2. [§2] §2 (or the section stating the main theorems): the application of the 2004 essential-radius bounds is invoked directly on the killed kernels, but without an explicit check that the singular potentials do not destroy the Feller property (possible when V is unbounded below), the spectral-radius estimates rest on an unverified hypothesis.
minor comments (2)
  1. Notation for the killed semigroup and the exponential functional of V should be introduced with a single consistent definition early in the paper rather than piecemeal.
  2. The list of processes in the abstract would benefit from a short table summarizing the precise assumptions on the driving processes and on V for each case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to confirm applicability of the cited theorems to the killed kernels. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the claim that the analysis 'relies on' the 2020 Perron-Frobenius theorem presupposes that the killed Feynman-Kac operators with singular V remain Feller kernels for all listed processes; no separate verification or additional assumptions ensuring continuity preservation after multiplication by the exponential functional of V are supplied, rendering the theorem's applicability unconfirmed and load-bearing for the compactness and long-time claims.

    Authors: We agree that the 2020 Perron-Frobenius theorem requires the killed kernels to be Feller. The manuscript states that the analysis relies on this theorem for the listed processes but does not supply a separate verification that singular V preserves the Feller property after the Feynman-Kac transformation. This is a substantive point. In revision we will add a dedicated paragraph (or short subsection) in §2 stating the conditions on V (e.g., Kato-class or local integrability assumptions standard for Schrödinger operators) under which the killed semigroup remains Feller for Lévy processes, kinetic Langevin dynamics, and interacting particle systems, thereby confirming applicability of the theorem. revision: yes

  2. Referee: [§2] §2 (or the section stating the main theorems): the application of the 2004 essential-radius bounds is invoked directly on the killed kernels, but without an explicit check that the singular potentials do not destroy the Feller property (possible when V is unbounded below), the spectral-radius estimates rest on an unverified hypothesis.

    Authors: We concur that the 2004 essential-radius bounds presuppose a Feller kernel. The manuscript applies these bounds to the killed kernels without an explicit check that V unbounded below cannot destroy the Feller property. We will revise §2 to include a brief lemma or remark verifying preservation of the Feller property under the paper’s standing assumptions on the processes and on V, thereby removing the unverified hypothesis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; relies on independently published prior theorems

full rationale

The paper's abstract states that the analysis relies on the 2020 Guillin-Nectoux-Wu Perron-Frobenius theorem for Feller kernels and the 2004 Wu essential spectral radius bounds. These are separately published external results (JEMS 2020 and PTRF 2004) whose authors overlap with the present work but which predate it and stand as independent contributions. No equations or steps in the provided text reduce any claimed prediction or compactness result to a self-definition, a fitted input renamed as output, or a closed self-citation chain within this manuscript. The central claims concern application of the cited theorems to new processes and singular potentials; any verification of Feller preservation is external to the cited theorems themselves and does not create a definitional loop. This is the normal case of building on prior literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper adds no new free parameters or invented entities; it invokes two external theorems as background.

axioms (2)
  • domain assumption Perron-Frobenius type theorem for Feller kernels (Guillin, Nectoux, Wu 2020 JEMS)
    Invoked to obtain compactness and principal eigen-elements for the killed semigroups.
  • standard math Bounds on essential spectral radius of bounded nonnegative kernels (Wu 2004 PTRF)
    Used to control the spectral gap after compactness is established.

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