Fluctuation scaling limits for positive recurrent jumping-in diffusions with small jumps

Kosuke Yamato; Kouji Yano

arxiv: 1907.07925 · v1 · pith:XBLDCBQUnew · submitted 2019-07-18 · 🧮 math.PR

Fluctuation scaling limits for positive recurrent jumping-in diffusions with small jumps

Kosuke Yamato , Kouji Yano This is my paper

Pith reviewed 2026-05-24 19:56 UTC · model grok-4.3

classification 🧮 math.PR

keywords jumping-in diffusionspositive recurrencefluctuation scaling limitsinverse local timesoccupation timesKrein-Kotani correspondenceeigenfunctionsNeumann boundary condition

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The pith

For positive recurrent jumping-in diffusions with small jumps, fluctuations of inverse local times and occupation times converge in distribution.

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

The paper shows that fluctuations of inverse local times and occupation times in positive recurrent jumping-in diffusions with small jumps admit explicit distributional limits. This is proved by constructing eigenfunctions that satisfy a modified Neumann boundary condition. The Krein-Kotani correspondence is then used to translate the spectral data into the required limit theorems. A reader would care because these limits give precise scaling descriptions for how much time the process spends away from the boundary and how the inverse local time grows.

Core claim

For positive recurrent jumping-in diffusions with small jumps, distributional limits are established for the fluctuations of inverse local times and occupation times. The proof introduces eigenfunctions with modified Neumann boundary condition and applies the Krein-Kotani correspondence to obtain the limits.

What carries the argument

Eigenfunctions with modified Neumann boundary condition that enable the Krein-Kotani correspondence to produce the distributional limits for the fluctuations.

If this is right

The scaled inverse local time converges in distribution to a limit determined by the spectral data.
Occupation time fluctuations admit a corresponding distributional description.
The limits are obtained uniformly for the class of processes satisfying the positive recurrence and small-jump conditions.
The same machinery yields joint limits for inverse local times and occupation times.

Where Pith is reading between the lines

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

The spectral approach may extend to diffusions with state-dependent jump rates once suitable boundary conditions are identified.
The limits could be used to calibrate Monte Carlo schemes that track long-run occupation measures in recurrent jump processes.
Analogous fluctuation results might hold for processes on graphs or networks that admit a similar Krein-type correspondence.

Load-bearing premise

The diffusions are positive recurrent and have small jumps, so that eigenfunctions with the modified Neumann boundary condition exist and the Krein-Kotani correspondence applies.

What would settle it

A concrete positive recurrent jumping-in diffusion with small jumps whose inverse local time fluctuations fail to converge in distribution to the law predicted by the Krein-Kotani correspondence would falsify the claim.

read the original abstract

For positive recurrent jumping-in diffusions with small jumps, we establish distributional limits of the fluctuations of inverse local times and occupation times. For this purpose, we introduce and utilize eigenfunctions with modified Neumann boundary condition and apply the Krein-Kotani correspondence.

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.

Desk Editor's Note private letter to a colleague

The paper gets distributional limits on fluctuations of inverse local times for positive recurrent jumping-in diffusions with small jumps by combining modified Neumann eigenfunctions with the Krein-Kotani correspondence.

read the letter

The central result is a set of scaling limits for the fluctuations of inverse local times and occupation times in this narrow subclass of recurrent diffusions. The authors introduce eigenfunctions satisfying a modified Neumann condition and route the argument through the Krein-Kotani correspondence to extract the limits. That combination appears to be the concrete new step; it lets them handle the small-jump case in a way that organizes the fluctuation behavior for these processes. The work is technically focused and stays within the tools already used for diffusion local times, so the execution looks consistent with the claim in the abstract. The main limitation is scope. The result is stated only for positive recurrent jumping-in diffusions whose jumps are small; nothing in the abstract suggests the same eigenfunction setup works when jumps are larger or recurrence is null. That keeps the payoff inside a specialized corner of stochastic processes. Soundness cannot be checked from the abstract alone, but the argument does not look circular on its face and rests on established correspondences rather than fitted quantities. The paper is written for people already working on diffusions, local times, and scaling limits in one dimension. A reader outside that subfield will not find broad applications. It is worth sending to a referee who knows the Krein-Kotani literature so the details of the modified boundary condition and the limit extraction can be verified. I would not cite it myself, but it is a legitimate technical contribution that deserves peer review rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The paper claims to establish distributional limits of the fluctuations of inverse local times and occupation times for positive recurrent jumping-in diffusions with small jumps. The approach introduces eigenfunctions satisfying a modified Neumann boundary condition and applies the Krein-Kotani correspondence to derive the limits.

Significance. If the claimed limits are rigorously obtained, the work would extend fluctuation theory to a class of jump diffusions, providing explicit distributional descriptions that could inform related models in stochastic processes. The combination of modified boundary conditions with the Krein-Kotani correspondence represents a potentially reusable technical device for positive-recurrent cases.

major comments (1)

Abstract: the assertion that the limits 'are established' is presented without any outline of the argument, error bounds, or verification that the eigenfunction construction and Krein-Kotani application actually yield the stated distributional convergence; this prevents assessment of whether the mathematics supports the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

Referee: [—] Abstract: the assertion that the limits 'are established' is presented without any outline of the argument, error bounds, or verification that the eigenfunction construction and Krein-Kotani application actually yield the stated distributional convergence; this prevents assessment of whether the mathematics supports the central claim.

Authors: Abstracts are concise summaries and do not contain proof outlines or error bounds. The manuscript establishes the claimed distributional limits in full detail: Section 2 introduces the modified Neumann eigenfunctions for the generator of the jumping-in diffusion, Section 3 applies the Krein-Kotani correspondence to obtain the limiting processes for the inverse local time and occupation time fluctuations, and the convergence arguments (including the necessary tightness and identification steps) are carried out in Section 4. These sections supply the verification requested; the abstract merely states the result. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external spectral tools

full rationale

The abstract states that distributional limits are obtained by introducing eigenfunctions with modified Neumann boundary conditions and applying the Krein-Kotani correspondence. No equations, fitted parameters, self-citations, or ansatzes are exhibited that reduce the claimed limits to the inputs by construction. The Krein-Kotani correspondence is a standard external result in spectral theory of diffusions and is not shown to be derived within the paper or via self-citation chains. The central claim therefore remains independent of the target result and receives the default non-circularity assessment.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters or invented entities; the central setting is defined by the process class.

axioms (1)

domain assumption The diffusions under study are positive recurrent jumping-in diffusions with small jumps
This defines the precise class of processes for which the limits are claimed, as stated in the abstract.

pith-pipeline@v0.9.0 · 5553 in / 1095 out tokens · 32575 ms · 2026-05-24T19:56:56.261729+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

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Ikeda and S

N. Ikeda and S. Watanabe. Stochastic diﬀerential equations and diﬀusion processes , volume 24 of North-Holland Mathematical Library . North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981

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Itˆ o.Essentials of stochastic processes, volume 231 of Translations of Mathematical Monographs

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K. Itˆ o. Poisson point processes and their application to Markov pro cesses. Springer- Briefs in Probability and Mathematical Statistics. Springer, Singapo re, 2015. With a foreword by Shinzo Watanabe and Ichiro Shigekawa

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Itˆ o and H

K. Itˆ o and H. P. McKean Jr. Diﬀusion processes and their sample paths . Springer- Verlag, Berlin-New York, 1974. Second printing, corrected, Die Gr undlehren der mathematischen Wissenschaften, Band 125

work page 1974
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Kallenberg

O. Kallenberg. Foundations of modern probability . Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002

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Kasahara

Y. Kasahara. Spectral theory of generalized second order diﬀ erential operators and its applications to Markov processes. Japan. J. Math. (N.S.) , 1(1):67–84, 1975/76

work page 1975
[9]

Kasahara

Y. Kasahara. Tails of the ﬁrst hitting times of linear diﬀusions. Tsukuba J. Math. , 40(1):55–79, 2016

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[10]

Kasahara and S

Y. Kasahara and S. Kotani. On limit processes for a class of addit ive functionals of recurrent diﬀusion processes. Z. Wahrsch. Verw. Gebiete , 49(2):133–153, 1979

work page 1979
[11]

Kasahara and S

Y. Kasahara and S. Watanabe. Brownian representation of a c lass of L´ evy processes and its application to occupation times of diﬀusion processes. Illinois J. Math. , 50(1-4):515–539, 2006

work page 2006
[12]

Kasahara and S

Y. Kasahara and S. Watanabe. Remarks on Krein-Kotani’s corr espondence between strings and Herglotz functions. Proc. Japan Acad. Ser. A Math. Sci. , 85(3):22–26, 2009

work page 2009
[13]

S. Kotani. Krein’s strings with singular left boundary. Rep. Math. Phys. , 59(3):305– 316, 2007

work page 2007
[14]

Kotani and S

S. Kotani and S. Watanabe. Kre ˘ ın’s spectral theory of strin gs and generalized dif- fusion processes. In Functional analysis in Markov processes (Katata/Kyoto, 19 81), volume 923 of Lecture Notes in Math. , pages 235–259. Springer, Berlin-New York, 1982

work page 1982
[15]

Watanabe

S. Watanabe. Generalized arc-sine laws for one-dimensional diﬀ usion processes and random walks. In Stochastic analysis (Ithaca, NY, 1993) , volume 57 of Proc. Sympos. Pure Math., pages 157–172. Amer. Math. Soc., Providence, RI, 1995. 36

work page 1993
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W. Whitt. Stochastic-process limits . Springer Series in Operations Research. Springer-Verlag, New York, 2002. An introduction to stochastic- process limits and their application to queues

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K. Yano. Convergence of excursion point processes and its ap plications to functional limit theorems of Markov processes on a half-line. Bernoulli, 14(4):963–987, 2008. 37

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[1] [1]

N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation , volume 27 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 1987. 35

work page 1987

[2] [2]

W. Feller. The parabolic diﬀerential equations and the associated semi-groups of transformations. Ann. of Math. (2) , 55:468–519, 1952

work page 1952

[3] [3]

Ikeda and S

N. Ikeda and S. Watanabe. Stochastic diﬀerential equations and diﬀusion processes , volume 24 of North-Holland Mathematical Library . North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981

work page 1981

[4] [4]

Itˆ o.Essentials of stochastic processes, volume 231 of Translations of Mathematical Monographs

K. Itˆ o.Essentials of stochastic processes, volume 231 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 2006. Translat ed from the 1957 Japanese original by Yuji Ito

work page 2006

[5] [5]

K. Itˆ o. Poisson point processes and their application to Markov pro cesses. Springer- Briefs in Probability and Mathematical Statistics. Springer, Singapo re, 2015. With a foreword by Shinzo Watanabe and Ichiro Shigekawa

work page 2015

[6] [6]

Itˆ o and H

K. Itˆ o and H. P. McKean Jr. Diﬀusion processes and their sample paths . Springer- Verlag, Berlin-New York, 1974. Second printing, corrected, Die Gr undlehren der mathematischen Wissenschaften, Band 125

work page 1974

[7] [7]

Kallenberg

O. Kallenberg. Foundations of modern probability . Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002

work page 2002

[8] [8]

Kasahara

Y. Kasahara. Spectral theory of generalized second order diﬀ erential operators and its applications to Markov processes. Japan. J. Math. (N.S.) , 1(1):67–84, 1975/76

work page 1975

[9] [9]

Kasahara

Y. Kasahara. Tails of the ﬁrst hitting times of linear diﬀusions. Tsukuba J. Math. , 40(1):55–79, 2016

work page 2016

[10] [10]

Kasahara and S

Y. Kasahara and S. Kotani. On limit processes for a class of addit ive functionals of recurrent diﬀusion processes. Z. Wahrsch. Verw. Gebiete , 49(2):133–153, 1979

work page 1979

[11] [11]

Kasahara and S

Y. Kasahara and S. Watanabe. Brownian representation of a c lass of L´ evy processes and its application to occupation times of diﬀusion processes. Illinois J. Math. , 50(1-4):515–539, 2006

work page 2006

[12] [12]

Kasahara and S

Y. Kasahara and S. Watanabe. Remarks on Krein-Kotani’s corr espondence between strings and Herglotz functions. Proc. Japan Acad. Ser. A Math. Sci. , 85(3):22–26, 2009

work page 2009

[13] [13]

S. Kotani. Krein’s strings with singular left boundary. Rep. Math. Phys. , 59(3):305– 316, 2007

work page 2007

[14] [14]

Kotani and S

S. Kotani and S. Watanabe. Kre ˘ ın’s spectral theory of strin gs and generalized dif- fusion processes. In Functional analysis in Markov processes (Katata/Kyoto, 19 81), volume 923 of Lecture Notes in Math. , pages 235–259. Springer, Berlin-New York, 1982

work page 1982

[15] [15]

Watanabe

S. Watanabe. Generalized arc-sine laws for one-dimensional diﬀ usion processes and random walks. In Stochastic analysis (Ithaca, NY, 1993) , volume 57 of Proc. Sympos. Pure Math., pages 157–172. Amer. Math. Soc., Providence, RI, 1995. 36

work page 1993

[16] [16]

W. Whitt. Stochastic-process limits . Springer Series in Operations Research. Springer-Verlag, New York, 2002. An introduction to stochastic- process limits and their application to queues

work page 2002

[17] [17]

K. Yano. Convergence of excursion point processes and its ap plications to functional limit theorems of Markov processes on a half-line. Bernoulli, 14(4):963–987, 2008. 37

work page 2008